Theory of Flow Properties of Attapulgite Suspension in Water. A Method forDetermining the RelaxationTime Parameter βA. F. Gabrysh, Henry Eyring, Michie Shimizu, and Jeanette Asay Citation: Journal of Applied Physics 34, 261 (1963); doi: 10.1063/1.1702595 View online: http://dx.doi.org/10.1063/1.1702595 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A method for an accurate T 1 relaxation-time measurement compensating B 1 field inhomogeneity inmagnetic-resonance imaging J. Appl. Phys. 97, 10E107 (2005); 10.1063/1.1857393 NMR of Absorbed Systems. I. A Systematic Method of Analyzing NMR RelaxationTime Data for aContinuous Distribution of Nuclear Correlation Times J. Chem. Phys. 51, 5673 (1969); 10.1063/1.1671997 Restriction on the Form of RelaxationTime Distribution Functions for a Thermally Activated Process J. Chem. Phys. 36, 345 (1962); 10.1063/1.1732507 Flow Properties of Attapulgite Suspension in Water Trans. Soc. Rheol. 5, 67 (1961); 10.1122/1.548887 RelaxationTime Model for FreeRadical Concentration J. Chem. Phys. 26, 1210 (1957); 10.1063/1.1743495

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 160.36.178.25 On: Sat, 20 Dec 2014 05:03:44

JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 2 FEBRUARY 1963

Theory of Flow Properties of Attapulgite Suspension in Water. A Method for Determining the Relaxation-Time Parameter ~

A. F. GABRYSH, HENRY EYRING, MICHIE SHIMIZU, AND JEANETTE ASAY

Institutefor the Study of Rate Processes and Department of Metallurgy, University of Utah, Salt Lake City, Utah

(Received 17 August 1962)

In two previous papers the stress-strain-time relations with respect to variables of temperature, con­centration, and pH were given for attapulgite suspension in water, in terms of rheopexy (dilatancy) and thixotropy. Stress relaxation and script S-like curves showing both thixotropy and rheopexy were deter­mined at various temperatures. Yield-point viscosities for the rheopectic and thixotropic curves showed activation energies of about 1 and 2 kcal, respectively. In the present paper Eyring's rate theory of viscosity is briefly discussed in terms of the well known generalized-viscosity equation

, Xi(J. sinh-l(Jis 1]= ~ -----.-,

i~1 OIl (J,s

which is applied successfully to given flow curves. A new method for determining the parameter (J, is dis­cussed.

INTRODUCTION

T Wo previous papers gave experimental results concerned with the study of the flow properties

of attapulgite suspension in water.1 In the first paper2 the stress-strain-time relations

with respect to variables of temperature, concentration, and pH were given in terms of rheopexy and thixotropy for flow curves where the shear rate oS range was 0--703/ sec.

In . the second paper3 stress-relaxation experiments were conducted under different conditions on samples which were relaxed for various durations of time be­tween successive cyclic deformations. Effect of pH on viscosity showed an inversion at a pH~6.3. A plot of the yield-point viscosities vs the reciprocal of tem­perature for the rheopectic and thixotropic curves showed an activation energy of about 1 and 2 kcal, respectively.

Although it is probable that thixotropic and rheo­pectic phenomena were involved in what appeared to be solely a thixotropic or rheopectic flow curve there were cases where, as the shear rate increased, the flow curves looped over in a manner similar to the making of a script S. Both the thixotropic and rheopectic phe­nomena were clearly involved. A log-jam model was used to explain all of the above phenomenon in terms of fluidity [which depends on (1) shear rate, (2) time since oS became zero, and (3) amount of surface] as a function of free water available to lubricate the needle­like attapulgite particles in suspension.

1 Electron micrographs of particle size and crystal habit of attapulgite are given in references 2 and 3 for a nonsheared sample and for samples that have been removed from the shearing region after various cycles of shear. The separation of bundles into single needles and a progressive breakage of needles during shear are clearly indicated.

2 A. F. Gabrysh, Taikyue Ree, H. Eyring, Nola McKee, and 1. Cutler, Trans. Soc. Rheo!. 5, 67 (1961).

3 A. F. Gabrysh, H. Eyring, and 1. Cutler, J. Am. Ceram. Soc. 40, 334 (1962).

In this paper we present: (1) an alternate method for the determination of the parameter f3 used in the well known generalized viscosity equation4; and (2) the non-Newtonian viscosities of several flow curves of attapulgite suspensions are calculated from the gen­eralized viscosity equation, using the parameters ob­tained by the new method.

THEORY

1. Non-Newtonian Viscosity

Eyring4 derived an equation for non-Newtonian vis­cosity based on the theory of Rate Processes. His theory was later generalized4 .5 and successfully applied to various systems: Christiansen, Ryan, and Stevens6

applied it to napalm in kerosene, hydrated lime in water, and sodium carboxymethylcellulose in water; Hahn et al.7 applied it to lubricating oils under pres­sure; Maron and PierceS applied the generalized equa­tion to spherical particles; Dahlgren9 applied it to bentonite (plate-like structure) suspensions. The gen­eralized equation is here applied to the bundle-needle­like structurelO of attapulgite.

For the above systems the relationship between vis­cosity '1/ and the rate of shear oS is given the following

4 H. Eyring, J. Chern. Phys. 4, 283 (1936); S. Glasstone, K. Laidler, and H. Eyring, The Theory of Rate Processes (McGraw­Hill Book Company, Inc., New York, 1941), p. 483; T. Ree and H. Eyring, J. App!. Phys. 26, 793 (1955).

5 R. E. Powell and H. Eyring, Nature 154, 427 (1944). 6 E. B. Christiansen, N. W. Ryan, and W. E. Stevens, Am.

Inst. Chern. Engrs. ]. 1, 544 (1955). 7 S. ]. Hahn, H. Eyring, 1. Higuchi, and T. Ree, NLGI Spokes-

man, 22, 121 (1958). 8 S. H. Maron and P. E. Pierce, ]. Colloid Sci. 11, 80 (1956). 9 S. E. Dahlgren, J. Colloid Sci. 13, 151 (1958). 10 T. F. Bates, "Selected Electron Micrographs of Clay and

Other Fine-Grained Minerals," Penn. State Univ. Mineral Indus­tries Experimental Station Circular No. 51 (1958).

261

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 160.36.178.25 On: Sat, 20 Dec 2014 05:03:44

262 GABRYSH, EYRING, SHIMIZU, AND ASAY

equation4

(1)

the subscript i refers to a particular flow unit and the summation is the extension over all the flow units in the system. Xi and CXi are theoretical flow parameters. X i is the fractional area occupied by the ith flow unit on the shear_surface, and

(2)

{3i is the relaxation time of the ith flow unit

(3)

These three quantities depend on concentration and two of them CXi and (3i depend also on temperature. A is the distance a unit moves between equilibrium posi­tions, }d\3 is the cross-sectional area of a given flow unit, A1 is the distance between planes of a given kind of flow unit, and k' (=kT/hexp(-t:..F;*jRT) in Eyring's absolute rate theory,4 where t:..F/ is the free energy of activation for the ith flow unit) is a rate constant, for the flow process at zero stress, for passage of the unit over a potential energy barrier.

Considering the water-attapulgite system one can assume the presence of two types of flow units, namely, water, whose behavior is taken as Newtonian and the bundle-needle-like particles of attapulgite whose be­havior may be non-Newtonian. For these considera­tions, Eq. (1) takes the form

'I1(S) = (X!f31)/CX1+ (X2(32)/cx2(sinh-l(32s)/(32s). (4)

To apply this generalized viscosity equation, it is necessary to determine the molecular parameters X i/ CXi and (3i. In Eq. (4) a plot of viscosity vs sinh-1(32s/1l2s results in a straight line for proper value of {32. The straight line has the slope X~2/CX2 and the intercept with the "viscosity" ordinate reprysents the Newtonian

8

7

6 . "Exch,onge" '2 !! pomt

'u 4 /~\ ' . ! ' . ·m 3 (b)i . \ . \

2 80

(0)

0 0 8

f. 10-> dynes/em'

FIG. 1. (a) Experimental (line) and calculated (heavy solid dots) values for first-cycle flow curves at five temperatures; 20% by weight suspension; pH~8.2. (b) Plot of yield point as a func­tion of temperature showing "exchange point" temperature of about 45°C.

unit X1{31/CX1. The correct value of {32 is found largely by a method of trial and error.

Kim et al. have recently developed a system where these parameters are uniquely determined for a number of systemsY Their method has shortcomings also. An­other method for determining the parameter (3 is re­ported below.

2. Determination of Parameter ~12

For simplicity Eq. (4) is put into the form

'I1=A+ B(sinh-l{3s/{3s), (5)

where A=X!f31/CXl and B=X2{32/CX2. For any three values of viscosity r,l, '112, and '113 and corresponding shear rate values Sl, S2, and S3 from the experimental curves, Eq. (5) can be put in the form

K. (6)

In most practical cases IlS is largerl3 than 1 and, from calculus if X> 1,

sinh-IX =In[X + (X2+1)!] =In(2X)+11/X2- ....

Thus Eq. (6) can be approximated so that sinh-Ills =In(2{3s). Equation (6) is then rewritten in the form

K:;:'K' log(2(3S2)-sdsllog(2(3S1)

i3z/ Sl log (2{3S3) - S3/ SI log (2(3S2)

(X+l2)-a2X

8 I st Reversible Cycle

7

·m 3

2

o

Experiment - -1°C -- 12·C _.- 2S·C

---- 54°C •.•..• 70·C

• Theory

3

f • 10"' dynH/em l

r f

f l J f / ,

! /

(7)

(8)

8

FIG. 2. Experimental (line) and calculated (heavy solid dots) values for reversible flow curves after repeated cycling of the samples used for Fig. 1 (a). 20% suspension; pH~8.2.

11 W. K. Kim, N. Hirai, T. Ree, and H. Eyring, J. App!. Phys. 31,358 (1960).

12 This method of determining {3 was recently developed and used successfully by one of the writers (M.S.) to determine the parameter for the flow curves of grease samples-K. Liang, Shao­mu Ma, T. Ree, and H. Eyring, "Behavior of Polymers in Grease" (manuscript in preparation).

131£ X<l (Le., (3s<l) then, sinh-1X=X-!X./3+!3/4X5/5 - ... ~X and '1=A+B=constant: this is Newtonian flow.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 160.36.178.25 On: Sat, 20 Dec 2014 05:03:44

FLO W PRO PER TIE S 0 FAT TAP U L G I T E SUS PEN S ION I N W ATE R 263

TABLE 1. The effect of temperature on the "molecular" parameters (valu~s of c~nstants for Fig. 4) of the viscosity equation for attapulgite. The "upcurves" exclude the Yleld-pomt phenomenon.

First cycle Downcurve

X 1{31

T Upcurve <>1 X 2/<>2 (CC) Shear stress (f) = (Poises) (dyn/cm2)

1 1180+5.85& 0.30 572.00 12 1300+5.06& 1.15 416.20 26 1080+5.048 0.10 462.80 54 930+3.08& 0.08 266.00 70 950+2.64& 0.10 98.61

where X = log(2{J'~I), a2= S2;'~I, a3= '~3;'~1, loga2= l2, and loga3 = h Then

X = [1 2- k' (a2l3-a312)/k' (a2- a3) -1+ a2]. (9)

Thus the parameter {J is given by

{J= log-IX/2s l .

COMPARISON WITH EXPERIMENT

1. First-Cycle Flow Curves

(10)

The apparatus used in these experiments is described elsew herel4•

Viscosity is given by Eq. (1) for a system which contains two or more different types of non-Newtonian units of different relaxation times. [As shown above, when {J is so small that (sinh-l{Js/{Js~l) the correspond­ing viscosity term is reduced to Newtonian viscosity; i.e., 7}N= (X{J/a)N.] The parameter {J2 for the curves given in Figs. 1 and 2 were determined in the manner given above under "theory". The remaining parameters were obtained by plotting experimental values vs sinh-l{Js/{Js and extrapolating the straight line thus obtained to sinh-l{Js/{Js= O. The slope of the line is X 2{J2/ ct2 and the intercept of this line with the ordinate 1/ is XI{JI/al. When the values of 1/1, 1/2, and 1/3 in Eq. (6) are such that the denominator in Eq. (9) approaches a limiting value, as it does for the upcurves in this study, then (Ji -H,O and shear stress takes the form of a Bingham plastic model.

The values of shear stress j and the flow parameter deduced for the up and down, first-cycle flow curves at five temperatures, are summarized in Table I. These parameters reproduce the experimental data satis­factorily as may be seen from Fig. 1 (a). In the figure, the lines are the experimental curves; the points are the data calculated from the parameters given in Table I.

The wide-gap spread at s= 700 secl between the 26° and 54°e flow curves indicate a possible existence of an initial temperature-dependent flow unit. This is further evidenced by the yield points of the upcurves,

14 A. F. G~brysh, H. Eyring, Shao-mu Ma, and Kai Liang, Rev. Sci. Instr. 33, 670 (1962).

Reversible cycle X j {31

{32 <>1 X'/<>2 (32 (sec) (Poises) (dyn/cm2) (sec)

9.29 0.80 888.44 3.03 50.30 0.76 560.43 20.52 27.13 0.68 464.60 56.50

2762.00 0.60 402.87 132.31 3.6XlOlo 0.42 2487.94 8700.00

which are indicated by arrows, and which vary be­tween 1.4X103 and 2.5X103 dyn/cm2. A plot of yield point vs temperature [Fig. 1 (b) ] suggests that a mechanism "exchange-point" temperature exists for aqueous attapulgite at about 45°e and a shear force of about 2.6X103 dyn/cm2.

2. Reversible Flow Curves

The parameters for the reproducible nonhysteretic curve were determined using the procedure outlined above. The values are summarized in Table 1. These parameters reproduce (points) the experimental re­versible curves very well.

3. One-Cycle Reversibility

Ten successive cycles (first cycle given in Fig. 1 and the last cycle in Fig. 2) with a peak of s= 703 secl were needed to effect a complete crystalline breakage and with the disappearance of dilatancy for the sample at l°e. Theoretically it should be possible to realize all of this breakage in one cycle by going to a high enough shear rate. An idea of the shear rate required to perform this work is obtained by calculating points (1) for a continuing upcurve for the first cycle and (2) the reversible curve, up to a common point of inter­section, see Fig. 3. In this manner it is determined that a 20-wt % suspension at '" 1 °e should experience a stress j of 7854 dyn/ cm2 at a shear rate of about s= 1140 secl to produce complete crystalline breakage, with the disappearance of dilatancy, during the first cycle.

DISCUSSION

The results of this paper together with the results of two previous papers2 ,3 lead us to a fairly detailed model of the behavior of attapulgite under shear. In­dividual crystallites are held together to form bundles. The force holding them together is electrostatic and arises as positive ions pass from the crystallite into the surrounding solution leaving the crystallite nega­tively charged. Two crystallites are held 'together by the positive ions in the water attracting the negative charge on neighboring crystallites. As the temperature

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 160.36.178.25 On: Sat, 20 Dec 2014 05:03:44

264 GABRYSH, EYRING, SHIMIZU, AND ASAY

12

II

10

9

8

"tQ

7

:'6 ... u .. .!!5 oU)

4

3

2

Temp '-I·C A ~ .

/ , f·1180+5.855 ~ " : ," , , '

Upcurve:

Downcurve, ~ = 0.80 . a,

X2 = 888.44 ldynes/cm2l a2

.B:i<sec)· 3.03

- Experiment

• Theory

f X 10-5 dynes/em"

, , /,,// +

, , / i /' ,

/ ' / I

FIG. 3. Peak shear-stress and shear-rate values at which re­versibility theoretically (dashed line and heavy solid dots) would occur in one cycle.

is raised the bond between crystallites is weakened by the increase in osmotic pressure of the positive ions, thus pushing the neighboring crystallites apart. The result is that with rising temperature there are more bundles of crystallites with a corresponding decrease in their size. The more numerous bundles tend to oglomerate more successfully, forming long chains which bridge the solution.

The thick bundles existing at low temperatures form fewer but stronger bridges. A bridge can be broken by shear stresses in two ways. First the bundle can be snapped in two breaking the crystallites, or the crys­tallites can be disentangled by the shear. From Fig. 1 (b) we see that at low temperatures very few bridges are formed. As the temperature rises the bundles be­come smaller and more numerous resulting in more bridging with correspondingly greater forces required to shear the increased number of bridges. Above 45° the bundles forming bridges are small and breakage of the bridge occurs by pulling the bundles apart rather than by snapping the individual crystallites. This is because the increase in osmotic pressure with tempera­ture weakens the electrostatic bonds between crystal­lites while temperatures has no such effect on the bonds between the atoms in the crystal itself.

In Fig. 4 we see that the rate of shear required to make cycles, after the first one, hysteresis free de­creases from 0° to 45°C and then rise again. This is because bridges at low temperatures are made of thicker bundles and require greater shear rates to break the crystallites. Above 45° the shear rate must go up again to break the crystallites faster than the

..

18

11

16

15

14

13

12

II

10

9

8

21 )(

,-6 (.)

!5 .... 4

3

2

• Peak shear rate for one-cycle reversibility.

--1°C

-- 12°C

-'- 26'C

f X 10-3 dynes/cm2

FIG. 4. "Theoretic3;l". ~hear rates and :;hear stress f required to gain first-cycle reverslblhty on attapulgIte for given temperatures.

bundles are pulled apart. The force of pulling crystal­lites apart involves viscous drag so that with fast enough shear rates the pulling apart is less likely than the snapping of the crystallites in the then bridges present at high temperatures.

Qualitatively the various types of viscous behavior of attapulgite thus seems to be understandable in terms of the bundles, observed in an electron microscope, on which the crystallites are held together by electrostatic attraction between the desorbed positive ions and the negatively charged crystallites. This picture is, of course, in accord with the observed fact that at the isoelectric point where the negative charge on the crystallites is just neutralized the viscosity passes through a minimum.2 ,3

ACKNOWLEDG-MENTS

One of us (}. A.) is grateful for the privilege of taking part in an undergraduate Research Participa­tion program sponsored by the National Science Foun­dation: Appreciation is extended to Professor J. )II. Sugihara for his encouragement and interest in the program. The authors are appreciative of the financial

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 160.36.178.25 On: Sat, 20 Dec 2014 05:03:44

FLO W PRO PER TIE S 0 FAT TAP U L G I T E SUS PEN S ION I N W ATE R 265

support of this research program by the American Chemical Society and the Petroleum Research Fund under grant 7S3-C6.

The authors are grateful to Miss Nola McKee for

the many helpful calculations not directly connected (0 this work and for preparing the drawings and the manuscript. Sincere appreciation and thanks are ex­tended to M. H. Miles for his help in many ways.

JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 2 FEBRUARY 1963

Rayleigh and Stochastic Scattering of Ultrasonic Waves in Steel

EMMANUEL P. PAPADAKIS*

Nondestr1tctive Testing Branch, Watertown Arsenal Laboratories, Waterto'wn 72, Massachusetts

(Received 5 July 1962)

The ultrasonic attenuation in SAE 4150 steel has been measured from 5 to 100 Me at various stages in its heat treatment. The attenuation is changed by changes in grain diameter and anisotropy. The anisotropy of grains transformed to martensite is lower than pearlite, while martensite itself becomes less anisotropic upon tempering. The dependence of the attenuation upon frequency f, grain diameter D, and anisotropy J.I agrees well with the theory of Lifshits and Parkhomovskii, who predict Rayleigh scattering proportional to J.l2JYf4 for x> 27rD and another type prop3rtional to JL2Df2 for A < 27rD. The anisotropy factor JL2 is lower by a factor of 10 for hardened, tempered steel than for iron.

1. INTRODUCTION

EXPERIMENTAU-S and theoreticaF-ll work on the attenuation of ultrasonic waves in polycrystal­

line metals has shown that Rayleigh scattering12 occurs when the wavelength is much larger than the grain diameter. In this region the attenuation is of the form

(1)

where A is a coefficient involving the ultrasonic velocity, p. is the anisotropy factor for each metal crystallite, Dis an average diameter for the crystallites, and f is the ultrasonic frequency. When the wavelength becomes comparable to the grain diameter (at higher frequen­cies), the attenuation arises from a different type of scattering8,9 and becomes proportional to p.2 DP, so

a= Bp.2 D P for A <2nD. (2)

* Present address: Bell Telephone Laboratories, Allentown, Pennsylvania.

1 W. P. Mason and H. ]. McSkimin, ]. App!. Phys. 19, 940 (1948).

2 (a) L. G. Merkulov, Soviet Phys.-Tech. Phys. 1, 59 (1956) translation of J. Tech. Phys. (U.S.S.R.) 26, 1956; (b) L. G. Merku­lov,]. Tech. Phys. (U.S.S.R.) 27, (1957) [translation: Soviet Phys. -Tech. Phys. 2,1282 (1957).

3 K. Kamigaki, Sci. Repts. RITU, Tohoku Univ. A9, 48 (1957). 4 W. Roth, J. App!. Phys. 19,901 (1948). "E. P. Papadakis and E. L. Reed,]. App!. Phys. 32, 682 (1961). 6 E. P. Papadakis, Ultrasonic Attenuation in SAE 3140 and

4150 Steel, Watertown Arsenal Laboratories, WAL TR 143/31 (April 1959).

7 W. P. Mason and H. J. McSkimin, ]. Acoust. Soc. Am. 19, 464 (1947).

8 I. M. Lifshits and G. D. Parkhomovskii, Record of Kharkov State University 27, 25 (1948); I. M. Lifshits and G. D. Par­khomovskii, J. Exptl. Theoret. Phys. (U.S.S.R.) 20, 175 (1950).

9 H. B. Huntington, J. Acoust. Soc. Am. 22, 362 (1950). 10 A. B. Bhatia,]. Acoust. Soc. Am. 31,16 (1959); A. B. Bhatia

and R. A. Moore, J. Acoust. Soc. Am. 31, 1140 (1959). 11 E. P. Papadakis, ]. Acoust. Soc. Am. 33, 1616 (1961). 12 R. L. Roderick and R. TruelI, J. App!. Phys. 23, 267 (1952).

The division of A = 27rD between the regions is given by Lifshits and Parkhomovskii.8

It has been shown that the anisotropy factor p. can be changed in iron-nickel alloys5 and in hardened steelS through processes not involving a change in the average grain diameter D or in the grain size distribution. In the case of the alloys, the process for changing p. is the martensitic transformation occurring below room tem­perature. The fcc austenite grain, a single crystal, breaks up into small bcc martensite platelets upon cool­ing. The platelets form in 24 directions with respect to the axes of the original grain, so the elastic anisotropy of the material occupying the original grain volume is reduced by an averaging process over all the platelets. In the case of steel, the martensitic transformation occurs at an elevated temperature during the quench­ing. Carbon atoms are retained interstitially in rapidly quenched steel making the martensite platelets body­centered tetragonal instead of cubic. The value of p. for austenitic carbon steel cannot be compared because the attenuation measurements cannot be made at the elevated temperatures. At those temperatures the grains are growing, anyway, so D is not constant. However, after the quenching, D remains constant throughout the life of the steel including its tempering. Thus in steel, during tempering, the interstitial carbon comes out of the iron lattice as iron carbide particles, and the lattice relaxes to a bcc configuration. The cubic lattice has a lower jJ. than the tetragonal lattice, so each martensite platelet becomes more isotropic and contributes less to the elastic anisotropy of the material within the boun­daries of the original grain. The platelets themselves contribute as scattering centers but have less effect than the original grain volumes because of the minute platelet size.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 160.36.178.25 On: Sat, 20 Dec 2014 05:03:44