An experimental study of the conformation of polyiso-butylene in a hydrodynamic shear field

ARCHIEF

THESIS DIGEST

AN EXPERINTAL STUDY OF THE CONFORMATION

OF

POLYISOBUTYLENE IN A HYDRODYNAMIC SHEAR FIELD

by

F. Richard Cottrell

Submitted to the Department of Chemical Engineering at the Massachusetts Institute of Techrology in July 1968, in partial fulfillment of the requirements for the

degree of Doctor of Science.

Thesis Supervisors:

E. W. Merrill, Professor of Chemical Engineering

K. A. Smith, Assoäiate Professor of Chemical Engineering v.

SciepsIouwkUfld

Techni'sch2

Hogeschcoi

TABLE OF CONTENTS

Page

Literature Citations

_..

24

Figures and Tables Appended

ACKNOWLEDGMENT

This work was supported by the office of Naval ReSearch under Contract Nonr 3963 (10).

A. Introduction

B. Scppe of this ExpHthentatiori C. Theoretical Background D. Conformation Criteria .. 1 4 4. 6

E. Materials and Apparatus 7

'1.

Solvent... '. .-

₇

2. Polymer

. . . _{. 7}

3. Light Scattering Apparatus . . . .. . . 8

4. Viscometers

9.

F. Data Acquisition and Manipulation 9

1. Characterization .. ₉

2. Conformation Determination . 10

3 Intrinsic Viscosity Shear Thinning Determination . 11

G. Results and Discussions . 11

1. Characterization . II

2. OHentãtiOn Angle Determination .,

.. .,.

, ... _. 13

3. Extension. RatiO DeterminatiOn. 16

4. Shear Thinnig . 18

THESIS DIGEST

A. Introduction ,. ..

-The conformation or statistical shape ofrandorn coilirtg polymer iolecules in, dilute solutions which are subjected, to a hydrodyiamic shear stress has been the, subject of considerable theoretical interest.

for the past three decades. Interest originally developed as a

theoretical justification of data pbtained in birefringence and intrinsic viscosity measUrements.

Essentially all of the statistical shape information appearing in the literature Which is applicable to macromolecules under shear is of a theoretical nature. Birefri'ngene. tneasurements have given experimental value.s of the orientation of the deformed macromolécule in flow but direct experimental data on the magnitude of the deformation has not been reported until now.

The theories which cOncern the. shear deformation of' random coiling macromolecules can be sub-divided into two categories. The

irst categoky concerns the macromo'lecular models which treat the chain as a number of Statistical segments :having various properties depending on the complexity of the model. Iiithese models (45, 39, 69), classical hydrodynamics is used tq calculate the force inpaMed to any one segment which is. also acted upon by thermal effects or Brownian motion as well as by entropy rubber elacticity Equations of motion

are written for the indidual segz.ents and the statistical shape determined from the resulting probability distribution fUnction relatingthe distance between the. two chain ends. ' .

The second category models the macromolecule to an elastic sphere which is homogeneous and isotropic with respect to its elastic properties. Cerf (7), after the original development of Hailer (24), applied this model

to macromolecules subjected tolow shear rates. Iittle quantitative

informatIon can be obtained from this model bause of the requirement of relating the thacroscopic elastic sphere parameters to the :molecuiar parameters and because of restrictions of its validity to low shear rates.

-2-Both categories qualitatively agree as to the natre of the coil

conformation und'r flow cOnditions. :The originally spherical random coilin macromolecule, when subjected to a sithple.hear field, takes an ellipsoidal statistical shape. The motion o'f the coil can best be described as a tank tread rotation Of the segments where the elliptical tank tread conformation

maintains aconstant orientation in the shear field. An illustrative

example of this:behavior is shown in Figure 1 - 1.

-FIGURE 1 l. THEORETICAL MACROMOLECULAR CONFORMATION IN A SHEAR FIELD

In this study, a new experiment, based on the the oi'ies of light scattering, has been developed which has alloed both the determination of the deformed macromolecular orientation in the Sheak field as wel as the determination of the magnitude of the defôrmatiôn along its extended and compressed directiohs.

As uch, the results presented

here give the first set of exerimental values of the macromolecular

conformation.

-To further elucidate the nature of the macromolecular

deformation, a study of the effect of the shear rate On the intrinsic viScOsity was also undertaken.

-3-energy dissipated in a simple shear fiec1 because of the existence of independent solute particles.. Such particles Introduce additional

shearing forces d,penijig upon their size, shape, and orientation insolutiOn.

At low shear rates where the macromolecular coil is not deformect,. the intrinsic viscosity is independent

f the shear rate. With

increasing shear stresses, however, real chains become increasingly deformed and give rise to a shear dependence of the intrinsic viscosity.

Theoretical considerations of the low shear rate intrinsic viscosity (based. on first Newtonian sOlution viscosities) in terms of the

macromolecular models have enjoyed a considerable,arnount of success and have. been verified with a wealth of experimental, data in the

literature. Of importance here, hOwever, is. the shear dependence of

the intrinsic viscosity in that it is reflective of the macromolecular deformation.

Theoretical considerations have been given to this effect, primarily by both Peterlin (40,, 42) and Cerf (9) Peterlin, in treating the change

of hydrodynamic interaction with coil expansion, predicts an initial decrease in the intrinsic viscosity followed by an increase. On the

other hand, Cerf, in accounting for chain rigidity effects, predicts either a continual decrease of the intrinic vicosity with increasing

shear rates or the existence of an asymptotic value indicative of a second Newtonian region_ Both of these theories have enjoyed a limited amount

of verification in terms, of experimental results. Table 2

- 2 lists a

summary of experimental conditions under which shear thinning data have been obtained and. published in the literature. _{Figures 2 - 6 and} and 2 -, 7 graphically illustrate the results of these studies.

Although the analysis of intrinsic viscosity shear dependence data at the present time does not lead to unequivocal conformation information, it is hoped that the simultaneous determination of both conformation

-4-Scope of this Experimentation

In this experimental trk, the öonformation or statistical shape

of a typical random coiling polymer molecule was determined under a simple hydrodynamic shear field. The variables of: interest in this

-1 mvestigation and their ranges are the shear rate, y (0 to 600 sec the polymer concentration (0 02 to 0 08 g/dl), and the polymer

6

6.

molecular weight, M (10 x 10 to 17 x 10 ).

-The intrinsic viscosity shear dependence was a]so determined under equivalent experimental conditions.

Theoretical Background

It has been realized for sOme time (41, 45) that light scattering would allow a dirt determination of the sheared macromolecular conformation. The parameter from. which conformation information

is determined is the inverse of the interference function, P01, which

can be written as:

2.2

2

P = 1 + l6rr

sin 0/2 p

+

&

where 0 is the observation angle, X the Wave length in the scattering media, and p is the mean squared radius of gyration of the

intramolecular distances projected or thes (a vector) direôtion. This direction is defined by a init vector a along the direction of

observation such that: -

--(1 - 2)

The physical significance of the interference function, F0, is that it is representative of the experimentally observed light scattered from: the polymer solution as a ratio to that which it would exhibit in the absence of destructive interference of the light scattered from

different segments within the macromo1iie.

The properties of the interference function allow a direct determination of the orientation angle, x, of the macromolécule.. This angle is defined here .s the angle between the positive dirtion. of the shear rate. and the major axis of the macromolecule. As

dictated by Equation. 1 -1, when s corresponds to the major axis Of the macromolecule, p takes on a mairnum value and P0'

becomes a minimum. That is, when s corresponds to the elongated axis of the macromolecule, one observes maximuni destructive interference or a minimum of the scattered intensity for a given observation angle., 6. The converse is true for orientation of the swith respect to the macromolecular minor axis. A new angle,

is then defined as the angle which s makes with respect to the

direction of the positive shear rate. Experimental measurements

of the scattered intensity versus the angle at cOnstant observation

angle, 6, and at constant shear rate., ?, Will then yield an

orientation angle, )(, from the minimum or maximum in this curve. Once the orientation angle is determined, the Value of p2 can be directly determined through the use of Equation 1 - 1. Values of the interference 'function are determined at decreasing values of the obsei'vation angle1 6, allowing a determination of p2 from the

limiting slope of this data. The experiment is carried out, however, at a constant value of .. An extension ratio can be defined as:

2 2 1/2

/ PC)

-5-(1 - 3)

where the subscript o refers to the zero shear case. This value of

is then taken to be an extension ratio of the statistical shape

S

-along the direction of s.

Statistical shape informatiofi of the macroipolecule can be defined only in the plane containing the s. In this study, this plane is the same

-6-as that defined by the dIrection of floW and the shear rate. Because projections of.the macromolecular dimensions are measured iflthis

plane, oily to measurements are independent. The relationship

relating these measurements is:

2

22.

2.2

= (s1 /s) a

+

.fs) b

where

s is the projection of s on the. major axis, 2 the. projection Of s on the minor axis, a the majOr axis extension ratio, and b the minor axis extensiOn ratio.

D. CoMormation Criteria.

The conformation of random coiling polymers, presupposed to be spherical in form, is dictated by either the root nean squared end-to-end distance, (r2)1/2, or the root mean squared radius of gyration

p2., The two are related by the relationship: (r2yV2 _=(p %6)l1

(1 - 5)

These relationships may be. obtained from either intrinsic viscoity or light scattering measurements. Along parallel lines, the mean squared eid-to-end distance along any linear direction, x2, can be used as a characterization parari-e.ter. This quantity is related to the radial end-to-end distance by the relationship:

/3

and the equivalent of Equation 1 - 5 can then be written as:

2 2

x '=PX6

where is the mean, squared radius of gyration of the projections

(1.- 6)

2 2

2_. + +

:2

2

.2

a b C.

-7-of the macromolecular dimension onto the x direction.

The conformation, Of the sheared macromoleclile is specified. by values of the orientation angle X of the macromolecule in the shear field as well as the extension ratios (or equivalently, the radii Of gyration. of Intramolecular distance projections) aloig the major axis, minor axis and axis perpendicular to the major and minor axes respectively. With the assumption that the distribution of the end-to-end distance around the mean remains Gaussian in

form along these axes, the statistical shape can be described _as

ellipsoidal. That is, the locus of points of equal probability of finding the termiial. segment at a distance x, y, and z from the

origin of a coordinate system at which the first segment is fixed is give1 by the relationship:

= constant _{(1 - 8)}

where a, b, and c are t1 extension ratios along the rnajor3 minor,

and perpendicular directions respectively. E. Materials and Apparatus

SolVent

Decahydro .napthaleñe (Fisher Chemical CO1 - purified grade) was used throughout this study. Its selection Was based on its

refractive index which could be varied by appropriate mixing of its cis and trans isomers to match the refractive index of the glass used in fabrication of the sample shear cell. Pertinent physical properties

of ihe mixture at 25°C are: refractive index, n =l.,476; Viscosity,

= 2.48 cp. Polymer

Two samples of pOlyisobutylene, designated by the

manufacturer, Enja3r Chemical Co., as L300 and L200 experimental samples, were used in this experimentation.. -The samples..were...

-8-fractionated by fractional precipitation from a dilute solution of benzene by the addition of the :riOn-solvent acetone. Approximately seven fractions of equal weight were obtained from each sample.

3. Light Scattering Apparatus

The lightscattering apparatus tied throughout this

experimentation was a modification Of the Sofica Light Scattering Photometer*.

Major modifications of the apparatus consisted of: Substitution of a coaxial cylinder device for the

standard sample cell. Both the rotatable inner

-cylinder (for example: OD = 0.892 inches)and

the stationary outer qylinder (for example: ID 0. 997

inches) were fabricated from Corning 7740 glass

20 _{1.474). Design of the apparatus allowed:}

.1. rotation of the inner cylinder at incremental

speeds to 600 rpm;

2. "run outs measured to

be less than 0. 0005 inches measured at the bob

extremity; 3. simple building block type construction; and 4.. reiioval of opticalparts for cleaning.

Modification of the incident beam so that ft: could be shifted. parallel to its standard propagation direction

in the plane perpendicular to the axis o rotation of

the shea.r cell. Displacement of.the incident beam to 043 inches of the. centerline of the shear cell could be

obtained. Reproducible adjustment of the incident beam to 0. 004 inches was possible. Incremental source.. apertures were also available to modify the dimension of the incidentbeam.

Modification of the secondary beam defining optics so that it could be oriented to observe any predetermined

point in the annular area of the shear cell. Incremental

apertures werealso available for this apparatus:. * Manufactured by:: Society Francaise

d. Introduction of an amplification system to allow accurate determination of both the seeondary intensity levels and differential readings of secondary intensity changes .upon

shearing the polymer solution.

'igure 4 -10 iUustr.tes a top view of the odified Sofica

instrument.

4. Viscômeters

The GDM (Couette type) viscometer (21) was used to obtain both the low shear rate, first Newtonian viscosities as well as the shear dependence of those viscosities (to 100 sec1)

A Ubbelohde viscometer was used to obtain the viscosity shear dependence under high shear rates. The dimensions ofthe Ubbelohde viscometer are: capillary radius, 0. 05 cm.; length,

25 cm; capillary reservoir volume. 0. 95 cm3.

_Pressirizing

the capillary reservoir allowed viscosity measurements to shear

3 -1

-.

rates..of 10 sec . Viscosities m this apparatus.were calculated from the wall shear stresses which were. calculated from the applied pressure readings and from the wall shear rates which were calculated according to Phillipoff et. al. (48). This latter method accounts for a. small non-Newtonian effect. Kinetic energy corrections were found to be negligible mthis apparatus

F. Data Acquisition and Manipulation

1. Characterization

Light scattering data were observed using both blue (X0 =. 4360 A°) and green (X = 5461 A°) Inc idenff irradiation which allowed

determinations of the sample weight averaged molecUlar weight and z-averaged mean squared radius of gyratioio These values were determined by the method of Zimm (L) High angle light scattering data were analyzed tO yield a easure of tie diSthibUliop of th,olcular weights as well s the number averaged mean squared end-to-end

-10-distance. These values- were determined according to the method

of Loucheux

Intrinsic viscosity measurements based on the shear independent,

first Newtonian viscosities were also determined for characterization

purposes. These data were observed on the GDM viscometér.

2. Conformation Determination

Experimental investigation of the sheared inacromolecular conformation was sub-divided into two

categoriese The first

-concerns the determination of the orientation of the macromolecule, while the second concerns the magnitude Of the deformation.

Orientation angle determination data consisted of determining the value of (mutual orientation of the s and direction of the

positive shear rate) at which the Interference function, P0, reached a maximum or a minimum. These measurements are indicative

of the orientation of the minor or major macromolecular axes in

the shear field respectively. Data were observed in this determination

at a constant value of the observation angle. Experimentally1 the

quantity P0 / P0 was calculated from observed values of the light

scattered from the shear cell-under zero shear conditions, 10

- p

as well as from the change of light scat-tered upon shearing the

solution, Al. The relationship used

in1this procedure was:

P L

-I

O,o P 0 &Ip = 1 -________

'u--i

_p 0 (1 - 9)

where P ,

_{0, 0} refers to the interference function valid for the zero

shear state (P_O,o constant, at 0 constant) and I-

o.

refers to the

light scattered from the solvent.

-Extensioi ratio data were obtained from the determination

of values of P /P

from the sheared state and from the1

o O,o -

-determination of values of P_o, from the zero shear state. - The. determination Of these data at decreasing incremental observation

angles, 0, allowed determinations Of values of p .through the use of

Equation 1 1. These data, however, were observed at a constant value of for a given hear rateJnlorder to yield an extension ratio valid for this direction of s with respect to the shear field. The major Portion of data acquired inths study was observed at =X or along the major axis. Minor axis extension. ratios were

determined from measurements at = x + 45° and the use of

Equation 1 - 4.

3. _{Iitrinsic Viscosity Shèar Thinning Determination}

Viscosity determinations of several sample concentrations

were developed by determination of shear st-rs shear rate. data

on the GDM viscometer to values, of approximately 100 sect

Viscosity determinations at higher shear rates (to 1O3 sec) were

determined from the Ubbelohde viscometer measurements. These values were based on the wall shear stresses and wall shear rates.

G. Results and DISCUSSiOnS 1. Characterizat ion

Table 1 - 1 illustrates typical results obtained in the characterization study.

'The simultaneous measurement of both the intrinsic viscosity and the. weight averaged molecular weight allowed _a determination of the constants of the Mark-Houwink equation, or:

= KMa

(1 - 10)

where K = 2.. 2 x l0 and a = 0.70. This relationship agreed Well with that Obtained by Ram (52) for the same solvent-polymer system.

12

-TABLE 1 H]. CHARACTERIZATION DATA

(*) For example L300-F3 signifies fractiOn number 3 takeh from L300 fractionation process.

(**) Based on first-Newtonian solution viscosities. (#) Z -Averaged root mafl, squared end-tp-end distance.

The ratio of Mw/M1N obtained from high angle light scattering

data yielded a measure of the efficiency of frat.Qn.tion.. As illustrated by the values of Table .1 - 1, the efficiency is considered excellent.

Reiatiqiship dictating the mean squared ënd-to-ejid distances of

the polymer coil to the niolecular weight 're also developed. Ue

of the number average o both of these parameters illuminated the

&d.erse nature of

the sample from this relationship and gave

a iI

which agrees with the. Fox-Floy (19) treatment of chain dimensions.

Values of. the. universal constant,

, were

also calculated from the intrinsic viscosity equation:

A value of foi the ftactiorated Cpecies yields = 1.46 t 0.16 x l0

Sample Orign(*

R()

dl/g

xlO A A° LS-3 L300-F3 25.0 16.8 1.1 7340 6030 LS-4 L300-F5 18.9 1Q5 1.3 620.0 :5400. LS-9 L300-F4 23.9 14.5 1.1 6900 6200 LS-2 L300 20.0 11.8 3.1 6470 3230

IS-i

L250 11.3 5. 9 .2 4840 2100

13

-This value agrees well With the theoretical predictions of Tschoegl (62)

or a large mo1ular weight species

_{immersed in a good solvent.}

It disagrees with the _{clasical1y accepted value of 2.1 x} io2l. In the characterization data, interference function data were. observed at both green (?.= 5461 A° _{and blue 4360 A°)}

incident irradjation _{Data obtained in this maimer Were found to}

Superimpose to a single _{curve by the use o the normalization}

.2

2

parameter (sin O//X

.

This result ngates the

_{existence of}

form depolarization .and _{illustrates the Debye assumption, concerning} the interference function to be correct.

2.

_{Qientation Angle Dterminatibn:}

The orientation angle was determined froth the value of

at which the value of P /P _{was observed to become a maximum}

- 0 0,o

Data for any one determination were observed at a constant Observation

I I

angle, 0, and shear rate,

_',

_{In that this value Of}

( max) is

indicative of. the orientation of the minor axis of the nacrornolecule with respect to the po5itive shear rate, the Orientation angle is calculated from.the relationship:

X max 9O

The determination of the _{position Of the maximum n the P0/P0}

versus _{curve was chosen over the alternative determination of}

the minimum be.cause Of the increased steepness exhibited _{by the} maximum.

Figur _{5 -14 and 5 - 15 illustrate data from which prieiitat ion} angle determinations were made. The method of determining _the maximum was aided sOmewhat _{by the fact that symmetry of data} existed around the thaximurn.

Figure 5 _{12 'illustrates the data Observed over the entire range} of values of _{and illustrates both the maximum afid the minimum.}

-14-Also shown by the, solidlineon this plot Is the solution to Feterlin's equation (45) for the interfetence function. This calculation is

scaled by both the. orientation and extenéion ratios observed in this study. The igniticance of the validity of the Peterlin method of solution' is that it Will allow the determinatioi of extension and. orientation information from the data exhibited by Figure 5 - l. This determination would require curve fitting but it would eliminate the tediou prOcedtire reqtiired.in exterision. ratio determination data acquisitions.

Figure 5 - 18 illustrates the summation of data obtained by the manner outlined above. AlsO shown in this figure is the

estimated limit of error in each determination. The dotted line

shown on this figure is the best fit of the theoretical euation:

tan'

(B /rn)

X=rr/4+

₂ (1 - 13)

where. a value of m= 3.6. Because of what appears to be a

discrepancy in form of the data to the curve, limited agreement may be obtained .by using values of m of 2. 7 to 4.5. Hence, the value of m Is Written as rn 3. 6

t

0. 9.

The normalization pãrametet, .B used here is a semI-theoretical parameter defined by the- relationship:

t

/c).'/

Iwj.'

i.12 (1 - 14)

where (Jc) is the viscosity number divided by the concentration,

is the solvent viscosity 2.48cp), and the other symbols take on their usual meanings. The macromolecular models predict as a normalization parameter the quantity B., or:

[Jiiwn

];5

-The quantity _{precipitates from the macromolecular models as a} normalizing parameter. to the distribution function for the distance

separating the chain ends.

The use of the parameter _{as a concentration normalizing}

parameter was originally determined through the use of

birefringence measurements.. The validity of its concentration

normalizing ability was also verified here _{The success of this} normalizing ability is thought to result from _{the representation}

within n/c of the additional shearing

_{fOrces applied to the}

macromolecules becuse of hydrodynamic doublet interaction.

It should be noted that the values of _{defined by quation 1 - 14} are. based on the first Newtonian viscosity number. _{Peterlin has} predicted the mutual dependence of both _{the. orientation angle and} the int±insic viscosity on the shear rate and the need of evaluating

the parameter _{utilizing the Viscosity number observed under}

the same applied shear field. _{This parameter, designated as}

/

cor, was calculated from the shear thinning

_{results reported here.}

Figure 5 - 19 illustrates a summation _{of' the orientation angle data}

plotted versus _{or. As exhibited by this figure and the d.shed}

theoretical curve, better _{agreement of the data tO the theoretical}

curve may be obtained in this relationship _{as versus that of}

Figure 5 - 18. The value of _{m from this curve, was determined to}

be: m = 2.5

0.5.

Table 5 - 9 lists the theoretical and _{experimental values of m}

obtained from the macromolecular _{models, birefringence data, and}

the results of this study. Comparison _{of experimental data is}

difficult bause of the existence of _{varying degrees of polydispersity.}

Polydispersity has the effect of lowering the experimental values of

in. _{Considering this factor, agreement to the experimental results}

from birefringence is considered good.

16

-models is difficult. As will be shown, large discrepancies exist in the comparison of the experimental and theoretical magnitudes of the extension of the macromolecule. As such, general agreement of the orientation angle results presented here to that predicted by theory is surprising.

3.. Extension Ratio Determination

The extension ratio was determined from the.square root of the ratio of the limiting slopes (c = constant, 0 o) of the

interference functions (data plotted according, to the method of Zimm) of the sheared to zero shear state.. Experimental data of 1° and

p

I observed for one concentration at a series of observation angles, 0, approaching zero can be applied as a corrective ratio to zero shear Zimm plot data. The data, however, are observed at a constant value, of

Figure 5 - 21 illustrates a Zimrn plot for the sample LS-9 which contains data valid for both the sheared state and zero shear state. These data were observed in the determination of the major axis extension ratios in that they were observed at values of _{'= x.}

Note that the extrapolation of both the sheared and zero sheared data at constant concentration to zero angle leads to an intercept valid for both sets of data. This fact, generally observed throughout this study, illustrates the negligible dependence of the second

virial coefficient on the shear rate.

The summation of data.valid for the major axis extension ratio is illustrated in Figure 5 - 26. Also shown, on this figure

are the results of a polyispersity analysis inwhich the monodisperse behavior was assumed; a Schultz-Zimm distribution of molecular weights was assumed; and the Z-averaged extension ratio calculated for a corresponding weight averaged value of . The averaging

technique 'was dictated by the physics governing the interference function and the experimental values of which were used.

17

-As illustrated by this plot, the parameter does appear to

normalize the concentration dependence of the extension ratios. However, a discrepancy does exist in comparing the data

illustrated by samples LS-9 and LS-3 with LS-4. Comparison with the results of the polydispersity analysis shows that this discrepancy can easily be explained in terms of vaiying

degrees.of polydispersity which exist, within any one sample. Assuming a Gaussian distribution of the probability

distribution function of distances separating the chain ends along the major and minor axes of the macromolecules, the

statistical shape takes on an elliptical form. Figure 5 - 27

illustrate the results Obtained in this experimentation in that the

solid lines represent the locus of points of equal probability of location of the terminal bead at a distance x, y, and .z from the origin at which point the first segment is fixed.

Figure 5 - 28 illustrates gi'aphicafly a comparison of _{the data}

exhibited in this study to that predicted by the macromolecular models. s. illustrated by this plot of the mean coil expansion

versus the parameter , a gtoss discrepancy, exists. _This

discrepancy is thought to be the result of coil rigidity effects which are not accounted for within the macromolecular models for which statistical shape data have been developed.

Consideration of the hydrodynamic forces acting on the

macromolecule will illustrate that macromolecules approximating

spherical symmetry rotate in the hydrodynamic shear field with

an angular Velocity of S'

/2 radians/sec. During the rotation, the

macromolecules experience what may be. described as a pulsatory radial force. For example, Figure 5 - 30 illustrates the

hydrodynamic force acting on any segment and illustrates the compression forces experienced by the segments in Quadrants II

and IV, while the coil experiences extension forces in Quadrants _{I and III.} Figure 5. 31 illustrates an example of the radial force which _{any line}

18

-of segments Would experience in its rotation around _{the center of mass} of the macromolecule.

The macromolecu].ar models, the results of which _{are discrepant}

with respect to the results reported here, treat the coil _conformation

as being dictated by an equilibrium of the viscous and elastic _forces.

-Thus, in essence, the molecule is supposed to respond instantly,

without phase lag, and maintain equilibrium with the forces

illustrated by Figure 5 - 31. _{However, a macromolecule with finite}

resistance to changes in coil conformation would not respond instantly and would exhibit a phase lag in its _{response to the force}

illustrated in Fure 5 - 31 and, in general, would not expand to the

equilibrium dimens ion dic.tated by a balance of its hydrodynamic and

elastic terins. _{It is just this effect which is thought to account for}

the discrepancies with respect to the macrOmolecular models observed here.

One other general effect: deserves mentioning. The hydrodynamic force applied to any one segment will be a direct function of the shear

stress. Thus, the dashed curve of Figure 5

- 31 is drawn to represent

the force a more viscous solvent would present to the segments With

respect to the solid curve at a given shear rate, it follows then:that:

the conformation of macromolecules which exhibit finite coil

rigidities will show higher coil extensions in a more viscous solvent at a given, shear stress than would be exhibited by the same

macromoleáules in a low viscOsity solvent. For example, in the limit of a very viscous solvent, high shear stresses (and high values

of ) can be obtained at very low values of the shear rate. As _such,

the freqtency of deformation of the coil will be low allowing an

equilibrium between the elastic and viscous forces, resulting in a

maximum deformation. 4. Shear Thinning

The viscometric studies resulted in viscosity determinations of

or: lim

C =0

19 -of approximately 1000 sec'.

Because of the normalization of conformation, data observed in this study, it was decided to use. this same procedure to normalize both the concentration dependence as well as the molecular weight dependence of the shear thinning behavior. For these purpOses, it is instructive to point out the following relationships:

TI fl

TJ/C

(1 -. 16)

TI' '

fc

= [TI]j

[I

(1 - 17)

where r10 is the. solvent viscosity, _{is the first Newtonian viscosity} of the sOlutiOn, TI is the solution viscosity at _Tlsp is the viscosity number at _.

is the. zro shear viscosity number, [j is the

intrinsic viscosity at , and [nj. is the intrinsic viscosity based on

first Newtonian values.

Figure 5 - 36 illustrates a plot of TI TI0 for the sample LS-:9 and TI_no.

illustrates the concentration normalization of these data which were observed from concentrations of 0.. 02 to 0. 08 g/dl. Figure 5 - 37 illustrates the data observed for various concentrations of the three fractionated samples and illustrates, within experimental error, the normalization offered to data of different molecular weights by the

parameter . The solid lines on this figure illustrate the results of

fitting the shear stress - shear rate data obtained at any one sample concentration tO a power law mathematical :form. and the extrapOlation of the data by mathematic al mãnipuiätion: of the power law equation to

20

-and that normalized by the _{method assure the validity of the.} normalition method at infinite dilution.

Figure 5 - 38 illustrates the data observed forthe

non-fractiOnated spies, LS-1 and LS-2. Some. of the apparent

scatter illustrated by these data can be resolved, by noting that higher concentration data fall significantly below that _{exhibited by} lower concentrations' of both samples. This behavior is illustrative of the limited concentration normalizing ability of the _parameter It is thought that the additional shear thinning observed, at higher concentrations is indicative of intramolecular entanglement effects.

Finally, comparison of the. data observed for the polydisperse samples to that exhibited by the fractionated species shows an apparent discrepancy in the normalized _{curves. That exhibited} by the unfractionated species.appé.àrsto be slightly displaced tO the right.?of thã exhibited by the fractionated species. This

discrèpancy:i not well understood. .

Comparison of the data exhibited here to that illustrated

Figures 2 - 6 and 2 - 7 which were published in previous

investigations illustrates almost complete agreement _{the data to} values of = 3. Beyond this point1 three general modes of behavior appear to exist. They are: first, continual intrinsic viscosity

decrease with increasing shear rates; second, the existence of a

seèond Newtonian region after an initial d'rease of the intrinsic viscosity; and third, the existence of an upturn of th intrinsic

viscosity after an original decrease. In general1 the first, two modes

of behavior are predicted by Cerf for coils having _{internal rigidity} effects which hinder coil expansion, while the. third node of behavior is predicted by Peterlin for coils having undergone appreciable

extension.

For reasons previously explained, coil expansion is _{expece.d. to} increase in a more viscous solvent because of the higher

_{Shear stress}

21

-frequency of the applied hydrodynamic

_{forces). As such, the upturn}

of the intrinsic viscosity is observed _{only in the most viscous of}

solvents as illustrated by the data of Burow in Figure 2 _{- 7.}

The existence of continual shear thihthng, as observed in this study and in the data reported by Wolff (66), appears to result from the.use of moderately viscous _{solventE. Thisbehavior is suggestive} of a limited coil expansion upon shearing.

The existence of a secOnd Newtonian _{region was illustrated by}

Go1ub22) for a polymer immersed _{in a low viscosity solvent. This}

behavior is suggestive of a limited asymptotic external shape of the macrornolecule.

H. Conclusions

Macromolecules deform into an oriented, ellipsoidal,

statistical shape in a sufficiently _{strong hydrodynamic shear}

field. The segments rotate

_{around the center of mass of the}

macromolecuie with an angular velocity and experience an

oscillatory radial force of twice _{the frequencyof rotation. The}

statistical shape, however, remains stationary in.a given shear

field. The angular velocity of the. coil is

_{' /2 radians / sec}

for coils approximating _{spherical symmetry.}

The deformation of the macroxnolecule s markedly less than that predicted by the theoretical

_{macromlecular models. This}

discrepancy is thought to be the _{result of resistances presented by} the coil to changes in coil _{conformation. Because of the oscillatory} nature of the deforming hydrodynamic shear forces acting upon the

polymer segments, equilibrium statistical _{shapes dictated by an}

equilibrium of the elastic and hydrodynaniic forces are not attained. The coil extension was found experimentally to adhere to the following properties:.

22

-conformation results observed here for both molecular Weight and concentratiOn.

b. The major axis extension.ratio for the system FIB in decalin yielded the normalized relationship valid for a

monodisperse species:

a = 1 + 0.013 0. 9 18)

4. The niacromolecular coil orientation was found to adhere to the following properties:

a The macromOlecule originally orientates its major axis at 45°tO both the positive velocity gradient and the direction of

flow. Increasing shear stresses causes the macromolecule to increasingly orientate from this position to the direction of flow..

The experimental behavior of the macromolecular orientation agrees generally with the results of the macromolecular models of Zimm (69).

t.

The parameter $ was found to normalize the experimental orientation angle data for both molecular weight and concentration.

The orientation angle can be given by the relationship:

,4tan1(

/3.6) (1 - 19) or: = /4 tan -1

(cor /2.5)

_{(1 - 20):}

vãi.id for the FIB - dealin system.

5. The normalization abilities of the parameter are thought to

be limited. Because of the oscillatory nature of the deforming hydrodynamic foxtes acting on any one segment due to coil rotation,

23

-the shear rate dictates a.frequency to which -the coil must adjust. The magnitude of the deforming force. is the hydrodynamic shear

stress.

Thus, appreciable changes of the solvent viscosity are expected to change the normalized behavior.

6. _{Shear thinning or the shear dependence of the intrinsic}

viscosity occurs as the macromolecule is deformed. The ratio of

the viscosity numbers from sheared to low shear state can be

normalized to a single curve when.plotted _{versus the parameter} . The normalized shear thinning curve obtained for PIB in

dec aim:

exhibited a slight dependency on sample polydispersity.; could not be correlated to a simple 2 dependency beyond values .of . > 1;

did not exhibit either a second Newtonian region or an upturn to values of 40;

agreed well with previous experimental results to values

of = 4.

7. _{In general, the shear thinning behavior observed at values of}

> 4 exhibits any one of the three following modes of behavior: A second Newtonian region exists which is suggestive of an asymptotic coil extension. This behavior is expected of macromolecules in low viscosity solvents (ri< 1 cp) where the coil expansion is severely limited.

Continual shear thinning exists which is indicative ofa continuous, limited, coil deformation. This behavior is expected of moderately viscous solvents (1 cp' rj 1 poise)

in which the coil deformation is limited.

An upturn in the viscosity ratio exists which is

indicative of appreciable coil expansion. This behavior is expected in only the most viscous of solvents (r ) 10 poise) where large coil deformations are possible.

24

-LITERATURE CITATIONS

Burow, S.P., A. Peterlin, and iD. T. Turner, Polymer, 2, 35 (1964).

Cerf, R., J. chim. phy., 48, 59 (1951).

9. Cerf, B., J. .Phys. Radium, L9.,, 122 (1958).

13. Copic, M., J. chim. phys., 54, 348 (1957).

Flory, P. J., ''Princip1es of Polymer Chemisfrya Cornell

University Press (1953).

Fox, T. G., Jr., J. C. Fox, and P. J. Flory, J. Amer. Chem. Soc.

I

24, 703 (1953).

Gilinson, P. J., C. R'. Douwalter, and E. W. Merrill, Trans. Soc.

Rheol., 7, 319 (1963).

Golub, M. A., J. chim phy., 54, 348 (1957). 24. Haller, W., Kolloid Z., 61, 26 (1932)..

36.

Loucheux, C., G. Weill, and H. Benoit, J. chim. phy., 55,

546 (1958).

Peterlin, A., J. Chem.,?.hys., 39, 224 (1963). Peterlln, A., 3. Chem. Phys., 33, 1799 (1960).

Peterlin, A., J. Poly. Sci., 23, 189 (1957).

Peterlin, A.,, Macromol. Chem., 44-46, 338(1961).

45.

Peterlin, A., W. Heller, and M. Nakagaki, J. Chem. Phys..,

28, .470 (1958).

48. Phillipoff, W., and F. H. Caskins, Trans.. Soc. Rheol., 2,

(1958).

Ram, A., Sc.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (1961).

.Shrman, L J., B. H. Sones, 'and L. H. Cragg, 3. App. Phy.,

24, 703 (1953).

62.

Tschoegl, N. W., J. Chem. Ihys., 40, 473 (1964).

66. IWoiff, C., J. chim..phy., 59, 413 (1962).

.Zimm, B. H., J. Chem. Phys., 16, 1100 (1948).

Zimm, B. H., J. Chem. Phys., 24, 269(1956).

TABLE 2 - 2..

LrrERATURE REVW

-TRINSIC viscosrry SHEAR

THNING

System Symbol Author Reference T C°

M6

[, xlO dl/g See Figure 2 - 6 PIB-Cyclohexa.ne

Fox, et. al. (20)

40 1.46 5,. 2 5 PIB-Carbontetra-. chloride " 40 1.46 4.6 5 PIB-Toluene 40 1.46 3.05 5 Polystyrene-Toluene

I

Sharman, et.a1.(5.) 1 25 2. 2.-4. 2;: 1.24 2.7.5 12 . 12 'I. 25 0. 288 1. 0 12 U . #. 15 2.2 4.2 12 U

I

. 45 2.2 4.2 12 It

.

-65 2..2 1.2 12 It I, 85 2.:2 4r2 . 12 Polystyrene-cyciohexane U 35 5. 2 1. 8 io 1 50, . 5.2 3.15 10

0

II 65' 5.2 3.6 .10 Polyisopreme-.

Golub ()

25 '4:;.. H 16. 2 20 benzene It It 25 2.55. 10.3 20 II It 25 3.4 12.8 20

Polystyrener

benz ene II

Polystyrene

-Pyralene 1499

= 1. 06 poise)

See Figure 2 - 7 Polystyrene

-Pyralene 1499 PMMA -Aroclor A x TABLE 2 - 2 (Cont) Copic (H) 20 1.06 3.65 40 I' 20 3..68. 8. 30 40 Wolff (66) 20 4.7 Wolff (66) 20 10.6 Burow (13) 15 2. 5 14.0

I0

[7]

[1L 0.I 0.I I I I I I I I I 1.0 /0

FIGURE 2 - 6 EXPERIMENTAL SHEAR DEPENDENCE OF THE

INTRINSIC

VISCOSITY.

3.0 J.0 0.!

100

FIGURE 2 - 7. EXPERIMENTAL SHEAR DEPENDENCE

OF THE INTRINSIC viscosrry.

See Table 2 - 2 for key.

Incident beam

Rotor

Inner wall - stationary cell

Secondary beam

Aperture cap _{Path of secondary} beam defining optics

FIGURE 4 - 10.. SCHEMATIC DIAGRAM

1.2 1.1

I.0

Fo.0,

0.8 0,7

Parameters required in caLculation

b=O.96

c1.0

a = 1.245 x

84°

/-,-'t( lIZ \r'?j7340A0

clockwlse rotation

£ counter clockwise rotation

I ii

4°

60 80 100 Experimental o = 90° y = 290 sec 0.036 gfdl 15.5 sample LS-3 /20 /40

.

160 /80

FIGURE 5 - 12. RATIO OF INTERFERENCE FUNCTIONS OF S1ARED TO ZERO

SHEARED STATES VERSUS ORNTATION OF SWrrH RESPECT TO THE SHEAR FIELD,

'.

SOLID LINE REPRESENTS SOLUTION TO

1.2 0.8 I. I 0.8 I _I

-Sample LS-4 c 0. 065 g/dl

0=90°

_-1

=58lsec

-\ o clockwise rotation ôounter clockwise. o rotation /20 /40 /60

/I8O

20 40

FIGURE 5 -.14. DATA EXAMPLE - ORJENTAI'ION ANGLE DETERMINATION 184 -I

oH

Sample LS-9-2

c = 0. 0688 g/dl..

0=90° 1

'=257sec

o clockwise 0 rotation counter 0 _clOckwise rotation

X+90°

L

I I I . I I 120 /40 /60 180 2b 40 60

FIGURE 5 - 15. DATA EXAMPLE - OR,NTATION ANGLE DETERMINATION

0

z

70

I

60

190 -50

tan(/3. 6)

= (n/cMw ii

y /RT T = 298°K, 2.48 cp

/0

20

30

I

40

-FIGURE 5 - 18. ONI'ATION ANGLE WI'H LIMrr OF ERROR

rsus

'

50

Symbol Sample Conc. 2

gfdl 40

rjsp/_dl/g LS -3 3.60 33. 3

A

LS -3 7.20 41.5 LS-3 6.67 48.5 LS -3 5.90 35.7

.

LS -4 6.50 27. 3 LS -4 5.20 21.0

£

LS -4 8.70 31. 0

is

6.88 36.0

i00

90 80 7o

.60

50 - 191 -20 0 1.0 / COR.

FIGURE 5 -.19. ORIENTATION ANGLE VERSUS 8 cor

LEGEND SAME AS FIGURE 5 - 18.

TABLES - 9. THEORETICAL-EXPERIMENTAL VALUES OFm 1.0 2.5 4.88 0.78 1.89. 3. 0. 9 2.StO.5 in -Authors Reference

Experimental System/ Model

Peterlin

dumbbell model

Zimm

(69)

free draining coil-Rouse model

Zimm

(69)

non-free draining coil-Rouse, model

Tsvetkov and Fris man

(63) PIB - gasoline Brodnyan, et.al. (4) :, FIB - decalin : This study

FIB - decalin (based on

This study

FIB - decalin (based on

'5

0-zero Shear v-N = 600rpm,, U = x 2D 3.0 4.0 2 it 3

sin O/2+Kc ,K

5x10

cd

'.5

0.9

Mw/Me 2

:w/1vr,=1. 33 Mw/M, 1.1 Symbol. Sample

0

ssumed monodispersed behavior 1.0+0.013 p0.9 Cone. 2

's

/c

g/dl xI0

dl/,

o

o

A

0 !0 20

FIGURE 5 - 26. EXTENSION RATIO ALONG MAJOR AXIS VERSUS

SOLID LINES REPRESENT THEORETICAL RESULTS OF _{POLYDISPERSITY ANALYSIS}

30 J_S-9 6.88 37.3 LS-9 4.58 32.6 LS-3 5.40 37.8 LS-4 6.50 27.3 LS-4 5.20 26.0

211

-FIGURE 5 - 27. STAT1TICAL SHAPE OF _DEFORMED

/1.0 /0.0 9.0 e,o 7,0

60

U 4.0 .2.0

I

- 212 Dumbbell mOdel

Free draining coil.

(Peterlin (39))

Non free. draining

-coil

(Peterlin U2)

Results of this study

\

,0r

.

.:I

Q 5O /011 /30

2

FIGURE 5 - 28. MEAN COIL EXPANSION VERSUS

..

COMPARON OF MACROMOLECULAR

217

-FIGURE 5 - .30. SC}MATIC DIAGRAM OF RESULTANT

HYDRODYNAMIC FORCES ACTING ON THE POLYMER COIL

FIGURE 5 - 31. TIME DEPENDENCE OF RADIAL FORCES.

40 I0

I

RIOUS

U'

lao Q LS

Di

-011? - 36.

to

7t-TL,10

-ji

0111

.,, 0) I I I I

II

I I

o LS-9

LS-3

0 LS-4

I I I I I I i I I I. I i I I I I I I

I,o1,

,10 40,

FIGURE 5 - 37. SIAR THrNNING BEHAVIOR OF FRACTIONATID

'q

it,

FIGURE 5 - 38. SHEAR THINNING

BEHAVIOR OF TIIE UNFRACTIONATED

SAMPLES I.' '__j I I I I I I I I, Symbol I 1 Sample I I I I I

Cpc.

g/ai

I I =

-iS-i

0.0956

-0 LS-I 0.0478

-0 LS.-1 0.0319 o LS-2 0.1008 A LS-2 0.0502 LS-2 0.0336 O LS-2 0. 0200 LS-2 0.0127 , ,

,,-ar