Detonation Sensitivity and Failure Diameter in Homogeneous CondensedMaterialsMarjorie W. Evans Citation: The Journal of Chemical Physics 36, 193 (1962); doi: 10.1063/1.1732296 View online: http://dx.doi.org/10.1063/1.1732296 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/36/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The propagation of detonation waves in non-ideal condensed-phase explosives confined by high sound-speed materials Phys. Fluids 25, 086102 (2013); 10.1063/1.4817069 Failure diameter of confined explosive rods AIP Conf. Proc. 505, 829 (2000); 10.1063/1.1303598 Detonation failure diameter studies of four liquid nitroalkanes J. Chem. Phys. 64, 2665 (1976); 10.1063/1.432520 Weak Detonations and Condensation Shocks J. Appl. Phys. 26, 969 (1955); 10.1063/1.1722147 Diameter Effect in Condensed Explosives. The Relation between Velocity and Radius of Curvature of theDetonation Wave J. Chem. Phys. 22, 1920 (1954); 10.1063/1.1739940

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ABSORPTION SPECTRUM OF AM3+ IN LaCI s 193

given J level with those of J levels having nearly the same energy.

An additional handicap to interpretation, pointed out by Sayre and Freed is that of working with a crystal of C3h symmetry and an ion of J=O ground stateY All of these complications combine to make a positive assignment of Land S very difficult. The only certain assignments for Eu+3, the rare earth analogue, are for the 5Do,1,2. These levels are of very low intensity and in

11 E. V. Sayre and S. Freed, J. Chern. Phys. 24, 1211 (1956).

THE JOURNAL OF CHEMICAL PHYSICS

the case of Am3+ shifted to the infrared.12 In the dilute crystals studied no levels were identified which could be assigned to the 5D multiplet. It would appear that pure crystals of Am salts would be needed to identify these levels. The levels observed probably belong to the 5 L, 5G, 5H, 51, 5 F multiplets but strong intermediate coupling prevents a simple direct correlation with the theoretical LS J values.

12 J. B. Gruber and J. G. Conway, J. Chern. Phys. 34, 632 (1961) .

VOLUME 36, NUMBER 1 JANUARY 1,1962

Detonation Sensitivity and Failure Diameter in Homogeneous Condensed Materials

MARJORIE W. EVANS

Stanford Research Institute, Menlo Park, California

(Received December 21, 1960)

A model is proposed for steady detonation waves in homogeneous condensed materials. The model is one in which the pressure profile in the direction of the wave motion is square between shock front and Chapman­Jouguet surface. The reaction rate in the steady zone is allowed to vary with the steady detonation wave velocity according to the temperature in the reaction zone. The model refines and makes quantitative an early suggestion by Eyring, Powell, Duffey, and Parlin. The results give a failure diameter, with dual-valued solutions at greater diameters. Predictions of failure diameter, wave velocity at failure diameter, induction time during initiation, and shock sensitivity for given charge dimension are possible. The predicted results for two materials of widely different detonation properties, liquid TNT and ammonium nitrate at theoretical crystalline density, are compared with observations.

THERE is general agreement that a steady detona­tion wave is well-represented by a Chapman­

Jouguet detonation wave which consists of three con­nected parts.I- 8 The first two are steady in a coordinate system which moves with the wave front. The first of these is a shock of short but finite rise time within which the pressure rises from initial pressure po to a value Pl. This compression causes a temperature rise from To to T I , which is sufficient to initiate a chemical reaction which continues into the second zone. There the chemical reaction continues until the Chapman­Jouguet surface is reached, at which position the pres­sure has a value p. which is less than PI and a tempera­ture T. which is greater than TI . The third zone is a

1 J. von Neumann, OSRD Rept. No. 549 (1942); ATL 159123. I Y. B. Zeldovich, Zhur. Eksptl. i. Teoret. Fiz. 10, 542 (1940);

translated in NACA Tech. Memo. 1261 (1950). 3 W. Doering, Ann. Physik 43, 421 (1943). 4 H. Eyring, R. E. Powell, G. H. Duffey, and R. B. Parlin,

Chern. Revs. 45, 69 (1949). 6 H. Jones, Proc. Roy. Soc. (London) A189, 415 (1947). 6 W. W. Wood and J. G. Kirkwood, J. Chern. Phys. 22, 1920

(1954). 7 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molectdar

Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954), pp. 797 fi.

g Marjorie W. Evans and C. M. Ablow, Chern. Revs. 61, 129 (1961) .

nonsteady rarefaction wave which adjusts the pressure and material velocity at the Chapman-Jouguet position to their boundary values.

As an approximation it can be assumed that the shock rise time is sufficiently short that none of the reaction occurs in the first zone. Furthermore, since the rarefaction wave causes the pressure, and thus the temperature, to decrease rapidly, it is often assumed that no chemical reaction occurs downstream of the Chapman-Jouguet position. On this model, then, all chemical reaction occurs in the second zone.

Many materials which detonate have been shown to have thermal decomposition rates corresponding to a single first-order rate process described by the equation

dE/dt=,,(l-E) exp( - Ea/ RT), (1)

where E is the fraction reacted, " is the frequency factor, and Ea is the activation energy. The temperature in a constant volume adiabatic reaction is related to the fraction reacted according to the equation

(2)

where T i is the initial temperature, Q is the energy released during the reaction, and c. is an appropriate average heat capacity. Substituting Eq. (2) into

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194 MARJORIE W. EVANS

Eq. (1) and integrating gives

t.=p-lj· (1-E')-1 exp[ ( EB, / Jdt', (3) o RTi+EQcv )

where t. is the time required for fraction E of the reaction to occur. Now according to Eq. (3), when E= 1 t.= 00.

However, for practical purposes reactions are assumed to go to completion in a finite time t.-l, which is com­monly approximated by the expression9,lO,n

t'_l= (c.RTNPEaQ) exp(Eo/RTi). (4)

According to Eq. (1) the rate of reaction dE/dt is initially small. It rises sharply to a maximum at a time approximated by Eq. (4) to give a value of E

close to 1 and a temperature close to the final tempera­ture. Finally the rate falls to zero and E approaches 1, both asymptotically. Thus in approximation the be­havior of E can be described as follows:

E=O for 0:5; t<t.=I, (5)

E=1 for t=t.=I.

In homogeneous materials for which Eqs. (1) to (5) are good approximations, one may expect that in the reaction zone of a steady detonation wave the tempera­ture will remain near the shock temperature T1, E

will remain near zero, and the pressure will remain near the shock pressure PI for a time approximated by Eq. (4), in which we now set Ti= T1. At time t.=1 the values of E, p, and T will change sharply to their Chapman­Jouguet values. On this model it is also reasonable to assume that the reaction is complete within a distance which is negligibly greater than the distance to the end of the steady zone of the wave.

Then, letting the subscript 2 designate values of pressure and temperature where the reaction is com­plete, the portion of the detonation wave between shock front and Chapman-Jouguet position is approxi­mately described as follows:

for

T=T2,

0:5; t <t._1,

for t= t.=I.

(6)

(7)

This model corresponds to that assumed as an approxi­mation by Shchelkin12 in a treatment of the instability of a plane detonation wave, and by Wood and Kirk­wood6 in obtaining an explicit solution for the relation between detonation velocity, reaction-zone length, and radius of curvature of detonation wave front.

Thus for steady detonation waves which are ade­quately described by this model the reaction is com­plete at a time t.=l given by Eq. (4) after a temperature

9 W. Jost, EXPlosion and Combustion Processes in Gases (McGraw-Hill Book Company, Inc., New York, 1946), p. 18.

l°H. W. Hubbard and M. H. Johnson, J. App!. Phys. 30,765 (1959).

11 J. Zinn and C. L. Mader, J. App!. Phys. 31, 323 (1960). I: K. I. Shchelkin, Zhur. Ekspt!. i. Teoret. Fiz. 36, 600 (1959)

[Translation: Soviet Phys.-JETP 9, 416 (1959)].

jump to T1. The time t.=1 may be in turn related uniquely to wave velocity U for any initial po and Po, where p is density, through the Hugoniot relation

(8)

a Hugoniot equation of state relating PI and PI for the unreacted material, and a thermodynamic relation giving Tl in terms of PI, PI, and the initial state. For the latter one may take the equation for temperature achieved in shock compression as derived by Walsh and Christian13

l 'l[h(T)e'Jr ] +exp( -bTl) --dT , (9) '0 c. lIugoniot

where T is the specific volume and

hH =!(dp/dT) (TO-T) +tp,

b=rpo,

r = KO/ {30P0c.,

{3= (1/T) (oT/aT)p,

K= - (1/T) (oT/aph.

The resulting relation between t.=1 and U is character­istic of the chemical and physical properties of the material and is independent of its dimensions. It may be expressed as t.=1 = fl (U /UO) , where UO is the velocity of a one-dimensional, steady-state, Chapman-Jouguet detonation (often referred to as the plane detonation), which is a constant for given material and given initial conditions. Alternatively 1.=1 can be related uniquely to PI to give a function t.=1 = h (pI), which is similarly characteristic of material properties and independent of dimensions. The function j2 describes the minimum pressure of square shocks of duration t.=1 which cause reaction in the material. For given shock strength PI, the corresponding value of t.=l is the induction time of the reaction. The function h may be viewed as a de­scription of the shock sensitivity of the material, since it describes the minimum pressure in an initiating square shock of specified duration that will build up to a steady detonation. Hubbard and Johnson10 calculated for a hypothetical propellant a shock sensitivity func­tion based on Eq. (4) and a single set of assumed material properties. Their function is expressed as t._l=j3(el), where el is the specific internal energy of the shocked material.

Now the velocity of a steady detonation wave in a charge of finite dimensions is dependent upon the dimensions. In particular the velocity of a steady detonation wave moving parallel to the axis of a cylindrical charge is dependent on the charge diameter as well as on the reaction time. The material flow between shock front and Chapman-Jouguet position

13 J.!M. Walsh:and R.:H. Christian, Phys. Rev. 97,1544 (1955).

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D E TON A T ION SEN SIT I V I T YIN CON DEN SED MAT E R I A L S 195

is a diverging flow. As a result the wave velocity is less than the theoretical wave velocity for a detonation wave in a charge of infinite diameter (the plane detona­tion wave velocity), in which the flow is parallel be­tween shock front and Chapman-Jouguet position. The amount of the divergence and hence of the velocity decrement in a charge of finite diameter is dependent upon the reaction-zone length, which is related to the reaction time, and upon the diameter. Approximate expressions relating the wave velocity U to the plane wave velocity uo, reaction zone length ~.=t, and the diameter D have been derived by several authors. Eyring, Powell, Duffey, and Parlin4 obtained

for charges without a casing, and

(UO-U) IU = 2.2~.=12poID(]'c

(10)

(11)

for charges with a casing, where (To is the mass per unit area of the case. Jones5 obtained

(U IUO)2= 1-3.2(~.=1/D)2

for charges without a casing, and

(ULU)IU=~12poID(]'c

(12)

(13)

for charges with a casing. Yet another expression has been derived by Wood and Kirkwood.6 This expression gives the velocity decrement in terms of the radius of curvature of the wave front s rather than in terms of the diameter. It thus cannot be used without knowledge or estimate of s. If observations of s exist it is likely to be more reliable than the expressions (10) to (13) as the derivation involves fewer a priori assumptions about the flow within the steady zone. Furthermore, it can be used for both cased and uncased charges. The approxi­mate expression cited by the authors as being a solution for a typical homogeneous condensed explosive is

(14)

where we have substituted ~'=l for ~* as given by Wood and Kirkwood.

Finally the reaction-zone width ~.=1 is related to the reaction time approximately according to

~.=l= 3tf=lU /4. (15)

Equation (15) is based on an approximate derivation first given by Eyring, Powell, Duffey, and Parlin4

[see also Eq. (5.2.23) of reference 8J with the added approximation based on detonation calculations for liquid and solid materials that Pol P20= l

The shock pressure on the curved shock front of a cylindrically symmetric detonation wave is a maximum at the axis and decreases with radial distance from the axis. The Hugoniot relation, Eq. (8), is valid on the axis, and in what follows we take PI and PI to be the values of shock pressure and density, respectively, on the axis. Since the wave is steady, its velocity U in the

10-4 L-~-L-._-L_~_--1. __ '--..-l 0.4 0.5 0.6 0.7 O.S 0.9

U!U'

1.0

FIG. 1. The function!l and selected functions g for liquid TNT.

direction of the axis is the same everywhere-within the steady zone.

Thus from an appropriate solution of the hydro­dynamic problem of the propagation of a steady detona­tion wave parallel to the axis of a cylindrical charge of finite diameter, one may obtain a function relating tf=! to U IUo with diameter D as a parameter. For ex­ample (15) may be combined with any of Eqs. (10) to (14) [assuming a knowledge of s if Eq. (14) is used] to obtain such a function 1._1 = g (U IUo, D).

An intersection of the curve II and any curve g for a given material at a given diameter is a solution for t._1 and U of the steady detonation wave at the given con­ditions. Equations (4), (8), (9), anyone of (10) to (14) or any alternative, and (15), together with a Hugoniot equation of state can be solved, with D as a parameter, for the values of U, PI, PI, TI, t.=1, and ~.=I, which represent steady detonation wave solutions. We will designate these solutions with a tilde over the symbol, thus 0, PI, PI, 1'1, l.=I, ~'=l. The typical form of the two functions it and g is shown in Fig. 1. For a diameter less than D, there is no solution. At diameters greater than D, the solution is dual-valued. It seems reasonable to associate the value DI with the well­known experimentally observed failure diameter. The physical interpretation of this result is that at a diameter less than the failure diameter a detonation wave cannot travel at a velocity sufficient to develop a temperat ure T1 such that chemical reaction can occur to develop a Chapman-Jouguet surface before rarefaction intervenes to quench all further reaction.

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196 MARJORIE W. EVANS

TABLE I. Assumed material properties.

Liquid TNT Ammonium nitrate

Po (atm)

To (OK) 354 300

Po (g/cm3) 1.472 1.725

c. (cal/g-OK) 0.35 0.41

r 1.25 1.50

Q (cal/g) 500 500

Ea (kcal/mole) 34.4 40.5

v (sec-I) 1011 •4 1013.8

UO (mm/ .usec) 6.6 6.0

Eyring, Powell, Duffey, and Parlin4 earlier obtained a similar solution for detonation wave properties as a function of diameter, with a minimum diameter and a dual-valued solution at greater diameters. Their solu­tions came about, as does the one proposed here, as a result of incorporating into their theory a reaction rate which varied exponentially with a temperature which was related to the wave velocity. However, in their treatment they suggested that the temperature which determines the rate of reaction is the Chapman­Jouguet temperature, rather than the shock tempera­ture as we suggest.

In order to make quantitative calculations which can be compared with experimental observations it is necessary to have chemical kinetic data over a suffi­cient range of temperature, and a Hugoniot equation of state over a sufficient range of pressure. The necessary data are available for only a few materials, and even in these cases the data must be extrapolated to reach the temperature and pressure regions of interest. To be most convincing a comparison between predic­tion and observation ought to be made for two ma­terials of closely similar detonation velocities but widely dissimilar failure diameters. Two such materials are liquid TNT and ammonium nitrate at theoretical crystalline density. For each of these materials there is some information on chemical kinetic constants, Hugoniot equations of state, and detonation char­acteristics. It is generally agreed that liquid explosives behave as homogeneous materials, and it is a reasonable assumption, one which is supported by observations of Campbell, Davis, and Travis14 on crystalline PETN, that single crystals behave as homogeneous materials with respect to detonation properties. We wish to make this assumption in our consideration of ammonium nitrate.

The calculated plane detonation velocity for liquid TNT is 6.6 mm/J.lsec, and for ammonium nitrate at crystalline density is 6.0 mm/J.lsec. The failure diam-

14 A. W. Campbell, W. C. Davis, and J. R. Travis, Phys. Fluids 4,498 (1961).

eters are widely different. The failure diameter of liquid TNT in Dural tubes at toO°C was estimated by Igel and Seely16 to be 12 mm. In contrast Amster, Noonan, and Bryan16 report that they were unable to detonate ammonium nitrate pressed to a density of 1.61 g/cm3 in a 3.65-cm diameter charge encased in a cold-rolled steel tube of 1.1-cm thickness. In this laboratory we have been unable to detonate ammonium nitrate pressed to a bulk density of 1.4 g/cm3 in charges of to-cm diameter encased in glass tubing. We believe that this is evidence that a single homogeneous crystal of ammonium nitrate would not be detonable at this diameter.

We have applied the above-described model to these two materials, with results described in the following section.

RESULTS

Table I shows assumed values for initial conditions po, To, and Po; thermodynamic properties Cv , r, and Q; chemical kinetic constants Ea and v; and ideal detona­tion velocity UO.

Q is estimated for both liquid TNT and ammonium nitrate at 500 caljg. The chemical kinetic constants for TNT are those obtained by Robertson17 over the tem­perature range 548-583 oK; those for ammonium nitrate are Robertson's values over the temperature 516--634°K.18 c. is assumed to be constant over the range of interest.

The ideal detonation velocity for liquid TNT is that cited by Garn.19 That for ammonium nitrate was calcu­lated according to the method described by Taylor.20

The Hugoniot equations of state which were used are shown on Fig. 2. That for liquid TNT is the one given by Garn.21 That for ammonium nitrate was obtained by D. R. Curran and B. O. Reese by the method, described in reference 22, which uses an explosive­produced oblique shock in a wedge-shaped specimen. The specimen was crystalline ammonium nitrate pressed to a density of 1.52 g/cm3• The measured points lay between 30 and 60 kbar. The curve has been extrapolated downward to crystalline density. It has been extrapolated to higher pressures by making use of a straight line extension of the shock velocity versus material velocity as suggested by Garn.21

The Grlineisen constant was estimated to be 1.25 for liquid TNT and 1.50 for ammonium nitrate from values of the constant for the limited number of

16 E. A. Igel and L. B. Seely, Jr., Second ONR Symposium on Detonation, February 9-11, 1955, pp. 321-335.

16 A. B. Amster, E. C. Noonan, and G. J. Bryan, ARS J. 30, 960 (1960).

17 A. J. B. Robertson, Trans. Faraday Soc. 44, 977 (1948). 18 A. J. B. Robertson, J. Soc. Chern. Ind. (London) 67, 221

(1948) . 19 W. B. Garn, J. Chern. Phys. 32, 653 (1960). :10 J. Taylor, Detonation in Condensed Explosives (Oxford

University Press, New York, 1952), Chap. VII. tl W. B. Garn, J. Chern. Phys. 30, 819 (1959). H S. Katz, D. G. Doran, and D. R. Curran, J. Api'\. Phys. 30,

568 (1959).

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DETONATION SENSITIVITY IN CONDENSED MATERIALS 197

300r-----,,-----.------~----~

250

200

LIQUID TNT:

'0=0.679 cm3/g

TO=354°K .NH4 NO,:

'0'0.580 em 3/g TO=300oK

j I 150 ",-

100

50

0'--__ -L __ -l. ___ ~..L:::::._ _ __..l

0.30 0.40 0.50 0.60 0.70

,,-tm3jg

FIG. 2. Hugoniot equations of state.

liquids and solids for which compressibility and co­efficient of thermal expansion data are available.

The relationships between shock pressure PI and wave velocity U, as obtained from Eq. (8) and the respective Hugoniot equations of state are shown in Fig. 3. The relationships between shock pressure PI and shock temperature T1, as obtained from Eq. (9) and the respective Hugoniot equations of state are shown in Fig. 4. The shock sensitivity functions f2=t._l(Pt) are shown in Fig. S.

The curves for Fig. 1 show the functions 11 and g for liquid TNT for the data given in Table I, with Eq. (14) as the diameter relationship. Experimental ob-

300 I i LIQUID TNT:

'0" 0.679 cm3jg

25') To" 354°K I'lH. NO,:

'0"0.S80cm3jg

200 To" 3000 K

~ .D ... 150 LIQUID TNT

0:'

100

50

O'--__ -l. ____ -l ____ -L ____ ~ ____ ~

4.0 4.5 5.0 5.5 6.0 6.5

u - mm/JLS6cond

FIG. 3. Shock pressure vs shock velocity.

<; ~

0.-

250

200

!50

100

50

LIQUID TNT:

500

To=0.679 cm 319 To= 354°K

"0= 0.580 cm 3/g To=300 o K

- LIQUID TNT

1000 1500 2000 2500 T,-OK

FIG. 4. Shock pressure vs shock temperature.

servations of the relationship between radius of 'curva­ture of detonation waves in liquid TNT ha~e been made by Igel and Seely.ls In the Dural tubes in which they measured a failure diameter of approximately 12 mm they observed a value for the ratio siD of ap­proximately 9 for tubes of diameter 31.8 and 63.5 mm. COOk23 observed siD ratios for various granular solid explosives and reported that the ratio varied from

,---.,....---,------.--,..--...,----, 10'

10

_-LIQUID TNT

10 5 10- 3 60 100 140 180 220 260 300

PI - kbar

FIG. 5. Reaction time vs shock pressure (the function 12).

23 M. A. Cook, The Science of High Explosives (Reinhold Publishing Corporation, New York, 1958), p. 99.

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198 MARJORIE W. EVANS

103 \

\ \ ~

I-

-1\ in z w 0

'" \ z I- 2

1012 j z 10

\ ... ~ 0

:; \ U) ,.

0 II: ::;

\ u

a: I-0

\ "" u. '" E 0

\ z .5 10 lOll r,<:t 0 \ z

II:

\ 0 u.

\ E E

LIQUID TNT --\

0

(510=9) 1010

\ \ / \. _I

10- 1 '--_--'-__ -'-__ ..L....._---''--_-' 109

0.6 0.7' 0.8 0.9 1.0 1.1 u/u·

FIG. 6. fj !UO vs diameter.

about 0.5 near failure diameter to 4 at large diameters. There is thus some uncertainty as to the value of siD. We have chosen to show the curves g in Fig. 1 calcu­lated assuming s/D=0.5, since we are interested in events at the failure diameter. The minimum diameter predicted in Fig. 1 is 4.2 mm. If siD is set equal to 9 the minimum diameter is found to be 0.24 mm. The Eyring equation (10) gives 0.6 mm and the Jones equation (12) gives 0.2 mm. The values obtained with the equations for cased charges, (11) and (13), depend on the value of (Fe' For reasonable values of (Fe one obtains minimum diameters in the range 0.004 to 0.02 mm. It is our belief that the Wood and Kirkwood equation, since it is based on fewer and less drastic simplifications, most nearly describes the actual flow and should by preference be used when values of s are available. In what follows we have used it.

In Fig. 6 the results for liquid TNT and ammonium nitrate are plotted as the ratio of the steady detonation wave velocity 0 to the plane velocity UO vs diameter D, assuming s/D=0.5. Also shown on this figure are the results for liquid TNT if s/D=9. From these graphs some comparisons of the predicted behavior with the experimental may now be made.

As to failure diameter, in the case of liquid TNT Igel and Seely have found that the failure diameter is probably near 12 mm. We predict (for s/D=0.5) that

the failure diameter is 4.2 mm. In the case of ammonium nitrate we have not been able to find any record, thoroughly substantiated, of a steady detonation wave at high bulk density. Thus in the sense that the model predicts a large failure diameter, 6600 km, the pre­diction is substantiated by observation.

Predictions may be compared with observations in a second way. The curve of Fig. 6 predicts that for liquid TNT failure will occur at a velocity of 6.25 mm/p.sec, that is at a velocity only 5% less than the plane wave velocity. This high value for Of/Uo, where Of is the steady detonation velocity at failure diameter, is char­acteristic of materials to which we might expect this model to apply, the homogeneous condensed materials. This observation has previously been made by Taylor.24

An example cited by Taylor is that of blasting gelatine at a density of 1.59 g/cm3• The plane detonation wave velocity is 8.00 mm/ p.sec, and the velocity at 3 mm, just greater than the failure diameter, is 7.60 mm/p.sec, a 5% decrease. In the case of liquid TNT, Igel and Seely15 quote a value for UO of 6.575 mm/p.sec and a value of 0 at D= 19.0 mm of 6.462 mm/p.sec. This is a decrease of 2% at a diameter which is believed by the authors to be slightly larger than failure diameter.

This behavior is not at all characteristic of granular explosives at relatively low loading density, for which the model described above cannot be appropriate. For such materials, the failure velocity is very much less than the plane velocity. For example, in this laboratory we have observed the steady detonation velocities of 100/5 ammonium perchlorate-PETN mixtures at a loading density of 1.3 g/cm3• The plane detonation wave velocity is 4 mm/ p.sec, while the detonation velocity at a diameter of 3 mm (in plastic tubing) is 0.8 mm/p.sec. This is an 80% decrease in velocity at a diameter which is an undetermined amount larger than the failure diameter. Similarly Taylor24 cites the example of granular ammonium nitrate at the low loading density of 1.0 g/cm3• The plane detonation velocity UO is 3.46 mm/p.sec at this loading density. At a diameter of 9 in., which is approximately the failure diameter for ammonium nitrate of typical coarseness, the velocity 0 was 1.4 mm/p.sec, or a de­crease of 60%. (The velocities and failure diameters depend upon the grain size of the ammonium nitrate, but the values cited are typical.)

Thus the predicted high value of O//Uo for homo­geneous condensed explosives would seem to be evi­dence supporting the validity of the model for such materials.

A third way in which predicted and observed results may be compared is with respect to induction times and shock sensitivities. The induction time is the time between the entrance of the initiating shock into the system and initiation of a steady detonation wave. (The velocity of the steady detonation wave with

24 Reference 20, pp. 142-143.

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DETONATION SENSITIVITY IN CONDENSED MATERIALS 199

respect to the observer will be the sum of the steady detonation velocity and the material flow in which the detonation wave is moving. It will initially be moving in the flow behind the nonreactive shock which pre­cedes it, so that the velocity with respect to the ob­server is higher than it is once the detonation wave has overtaken the nonreactive shock. This was pointed out by Chaiken25 and by Campbell, Davis, and Travis.14

The latter have made measurements of induction time, which they call t*, and the detonation velocity in the compressed material behind the initial shock wave, which they call D*, for several homogeneous explosives.) According to the model the induction time for a given entering shock is given by the appropriate curve of Fig. 5, where t.=l is the induction time to be expected for a square shock of intensity Pl. Campbell, Davis, and Travis14 measured induction times as a function of initial shock pressure for several liquid explosives. They list one set of preliminary data for liquid TNT, which shows an induction time of 0.7 JLsec for an initi­ating shock pressure of 125 kbar. From Fig. 5 the model predicts an induction time of 0.7 p.sec for a shock of 146-kbar strength and an induction time of 6.8 p.sec for a shock of 125-kbar strength. In view of the estima­tions involved in choosing values of r, Ea, v, and the extrapolation of the Hugoniot equation of state, this is believed to be fairly good agreement. It is perhaps worthy of note that Campbell, Davis, and Travis found in experiments with nitromethane that induction time can be artificially shortened if the wave entering into the receiver is not smooth.

The curves of Fig. 5 can be used to predict minimum shock sensitivity to a square shock. Since the pressures involved in shock initiation exceed the ultimate strength of materials, the entering rarefaction waves from the side boundaries may limit the duration of a square shock at the axis. Thus, if the diameter of booster and receiver are the same, say 20 mm, a square shock may not have a duration of greater than approximately (10 mm)/(3 mm/p.sec) =3.3 JLsec, where 3 mm/JLsec is an assumed value for the velocity of the inward moving rarefaction wave. Then according to Fig. 5, for liquid TNT any square shock of strength less than 132 kbar is incapable of initiating liquid TNT in that particular geometry. Similarly for a diameter of 30 mm, for which one estimates a shock duration of 5 p.sec, Fig. 5 pre­dicts that a shock strength of 128 kbar is the minimum necessary to initiate detonation. These predictions may be compared with some results of Garn.2l In his experi­ments a shock emerging from a cylindrical booster of 30-mm diameter after passing through an aluminum plate of 34-mm diameter passed into a cylinder of liquid TNT of 22-mm diameter. Garn reported that a shock of 110 kbar failed to initiate detonation, while a shock of 165 kbar initiated detonation. The initiation by a 125-kbar shock reported by Campbell, Davis and

15 R. F. Chaiken, J. Chern. Phys. 33. 760 (1960).

Travis as described in the previous paragraph was in a charge setup in which the diameter of the liquid TNT was at least 8 in. For such a diameter Fig. 5 predicts a shock strength of 125 kbar as being just sufficient to initiate (assuming a sound velocity of 3 mm/ p.sec). These results would seem to support the predictions from the model.

Finally the results predict two possible steady detonation waves at given diameter. The waves which are commonly observed lie on the high-velocity branch. The reason for this may have to do with the stability of the waves. The reaction zones for the high-velocity waves are narrower than those for the low-velocity waves. One might expect waves with narrow zones to be more stable, because for these waves the Chapman­Jouguet point on the axis is further from the source of pressure perturbations which may arise at the side boundaries. Perturbations resulting in a decrease in pressure within the reaction zone would decrease the temperature. For a detonation described by Eqs. (6) and (7) this could result in a quenching of reaction and of the detonation wave. We suggest that if at a given diameter the initiating shock pressure Pi has a value equal to the value of Pl for the lower velocity wave for that diameter, a steady detonation wave having a velocity corresponding to the appropriate point on the low velocity branch of the (j /uo vs D curve may be formed. For a cylinder of liquid TNT of 20-mm diam­eter, the s/D=0.5 curve of Fig. 6 predicts a velocity for this wave of 5.5 mm/p.sec, corresponding to a shock pressure of 170 kbar as given by Fig. 3. If the initiating shock pressure Pl is less than th for the low-velocity branch of the curve but nevertheless of sufficient dura­tion as determined from Fig. 5 to complete reaction, one might expect the reaction wave to build up to the steady wave solution on the low-velocity branch. This lower velocity wave would very likely have only a short lifetime, changing at the earliest favorable opportunity into the high-velocity steady wave.

Price and Jaffe26 have reported a result which may be an example of a steady wave on the low-velocity branch of the curve. They quote observations, originally reported by Majowicz and Jacobs,27 on cast TNT of initial density 1.58 g/cm3 for which the expected steady­state velocity, the one usually observed and that corre­sponding in our analysis to the high velocity branch, is 6.7-6.8 mm/p.sec. In an initiation experiment by the wedge technique the cast TNT exhibited a constant velocity of 5.23 mm/JLsec. Price and Jaffe cite another experiment measuring the shock strength resulting in a 50% probability of initiation of a detonation, in which a similar low velocity was observed by the continuous wire method.

We do not believe that the low-velocity branch is related to the low velocity persisting detonation waves

20 D. Price and 1. Jaffe, ARS J. 31, 595 (1961). ~ J. M. Majowicz and S. J. Jacobs, Bull. Am. Phys. Soc. 3.

293 (1958).

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200 MARJORIE W. EVANS

observed under certain circumstances in liquid and gelatinous explosives.28 •29 The observed velocities are much lower than those predicted by the model of this paper, lying in the range 1-3 mmjp.sec rather than the range 5-6.6 mmj p.sec as predicted by Fig. 6. These low velocities are characteristic of heterogeneous or granular explosives which are generally acknowledged to react according to the grain-burning mechanism originally proposed by Eyring, Powell, Duffey, and Parlin.4 The hypothesis that the low-velocity detonation waves in

J8 Reference 20, Chap. 10. 19 Reference 4, p. 135.

THE JOURNAL OF CHEMICAL PHYSICS

this instance are the result of reaction according to a grain-burning mechanism rather than according to thermal decomposition throughout a homogeneous medium is supported by the observation that these waves occur in aerated explosives, with the low velocity being favored when the bubbles are large.28

ACKNOWLEDGMENTS

The author takes pleasure in acknowledging com­ments and suggestions made by Drs. S. J. Jacobs and Donna Price of the U. S. Naval Ordnance Laboratory, White Oak, Maryland during the preparation of this paper.

VOLUME 36, NUMBER 1 JANUARY I, 1962

Microwave Spectrum, Structure, Quadrupole Interaction, Dipole Moment, and Bent C-CI Bonds in 1,1-dichlorocyc1opropane*

W. H. FLYGARE,t A. NARATH,t AND WILLIAM D. GWINN

Department of Chemistry, University of California, Berkeley 4, California

(Received June 12, 1961)

The microwave spectrum of CaH.CJ,35.35, CaH.CJ,35.37, CaD.Cb35.35 CaD.Cb35.37, CaHaDCJ,35.35, CaHaDCI235.a7 (cis and trans), and C2CI3H.CJ,35.35 have been investigated. Using the derived rota­tional constants, a complete structure for the molecule is calcu­lated. The structural parameters are dc-H = 1.085 A, dc(2)-C(a) = 1.534 A, dc(I)-C(2)=1.532 A, de-Cl=1.734 A, LH-C-H= 117°35', LC(2)-C(3)-H2=153°37'±15', LCl-C-Cl= 114°38'. K for C2ClaH.CJ,35.a5 was very close to zero (±O.OOOl), and it was not possible to determine its sign unambiguously. Using the negative value instead of the more likely positive value, a slightly different structure was obtained.

High-resolution measurements were carried out on the Cl

INTRODUCTION

I N earlier work on methylene chloride,l it was found that both the HCH angle and the CICCI angle

were greater than tetrahedral, both angles being about 112°. From this it was inferred that the C-Cl bond was probably bent. In this earlier work, the quadrupole­coupling constants were also measured, but only the diagonal elements of the quadrupole-coupling tensors in~the inertial axis system.were determined since the first-order theory was adequate to explain all of the experimental data. As a result, it was not possible to find the orientation of the principal axes of the quad-

• Based in part on the doctoral theses of A. Narath (1959) and W. F. Flygare (1961) University of California, Berkeley.

t G. E. Fellow in Chemistry, 1960-61, California Research Summer Fellow, 1960. Present address: Department of Chemis­try, University ofJlllinois, Urbana, Illinois.

t Present address: Sandia Corporation, Albuquerque, New Mexico.

I R. J. Myers and W. D. Gwinn, J. Chern. Phys. 20, 1420 (1952).

nuclear quadrupole hyperfine structure in order to obtain the complete quadrupole coupling constant tensor in the CaH.CJ,35.35 principal inertial axis system. The values obtained are x •• = -43.545 Me, xbb=4.100 Mc, xcc=39.445 Mc, all ±0.OO5 and Xab= -51.5±0.3 Me. The principal quadrupole-coupling con­stant tensor elements are Xa= -76.4±0.3 Me, xp=37.0±0.3 Mc, and Xcc=39.445±0.OO5 Me. The angle between the principal quadrupole axis component along the C-Cl bond and the C-Cl internuclear line is 0±15'. This is strong evidence against bent C-CI bonds in this compound.

Detailed analysis of the Stark effect in the presence of quadru­pole interaction leads to a dipole moment of 1.58 debye.

rupole tensors without making assumptions concerning the symmetry of the C-CI bonds.

In the attempt to obtain more information concerning these or similar C-Cl bonds, the work on 1,1-dichloro­cyclopropane was undertaken in the hopes of deter­mining the direction of the principal axes of the chlorine nuclear quadrupole-coupling tensors. The axes of the quadrupole tensor represent the orientation of the electron cloud about the CI atom. The angle between the C-Cl internuclear line and th(axes of the quad­rupole tensor should be a measure of the degree of bending in the carbon-chlorine bond.

The determination of the principal axes of the quad­rupole-coupling constant tensors in 1,1-dichlorocyclo­propane involves the determination of all the tensor elements in the principal inertial axis syst'em. Using perturbation theory to calculate the energy, the off­diagonal elements enter only in the second order.

Unfortunately for our present purpose, first-order theory is wholly adequate to explain all of our spectral

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