Polymers as Rheology Modifiers - American Chemical Society

Chapter 2

Rheological Measurements Robert K. Prud'homme

Downloaded by UNIV OF GUELPH LIBRARY on August 6, 2012 | http://pubs.acs.org Publication Date: May 13, 1991 | doi: 10.1021/bk-1991-0462.ch002

Department of Chemical Engineering, Princeton University, Princeton, NJ 08544

The goal of this chapter is to provide a working knowledge of basic rheological terms, experimental results, and simple rheological models. The discus sion is arranged in the following manner. In the first section the basic rheological flows and material functions (such as viscosity and dynamic moduli) are defined. In the second section the responses of general classes of polymeric fluids in these basic flow fields are presented. Emphasis is placed on the relationship between rheological response and material structure; that is "structure-property relationships" from a rheological point of view. In the final section simple rheological models are presented that can be used to represent the flow behavior of polymeric fluids. Flows and Material

Functions

Many i n d u s t r i a l processes subject f l u i d s to complex flow and temperature h i s t o r i e s . To understand these complex flows, the response of the f l u i d s i n simple flow f i e l d s are studied to determine t h e i r "material functions" such as the v i s c o s i t y , normal stress c o e f f i c i e n t s , or dynamic modul. In turn, these material functions are used to select appropriate mathematical models to describe the f l u i d rheology ( c a l l e d "constitutive equations" -- the subject of the f i n a l section) which can be used to predict the flow i n complex geometries and flow f i e l d s . In addition, the material functions can be used to characterize materials, for example the weight-average molecular weight of a polymer can be determined from zero-shear rate v i s c o s i t y measurements at high polymer concentrations. The d e f i n i t i o n s of several simple flows and associated material functions are given below. The measurement of material functions i n these flows defines the practice of "rheometry", and several s p e c i a l i z e d references are available that cover this area (1-5). 0097-6156/91/0462-0018$08.50A) © 1991 American Chemical Society

In Polymers as Rheology Modifiers; Schulz, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

2.

PRUD'HOMME

19

Rheological Measurements


Stress and Strain. The d e f i n i t i o n of stress requires the s p e c i f i c a t i o n of the d i r e c t i o n of the force and the o r i e n t a t i o n of the surface upon which the stress acts. S i m i l a r l y , the d e f i n i t i o n of the rate of deformation or v e l o c i t y gradient requires s p e c i f i c a t i o n of the d i r e c t i o n of the v e l o c i t y and the d i r e c t i o n i n which the v e l o c i t y v a r i e s . Figure 1 shows f i v e flow f i e l d s extensively used i n rheological measurements; steady shear, step shear rate, u n i a x i a l extension, b i a x i a l extension and o s c i l l a t o r y shear. Steady-Shear Flows. Consider the flow shown i n F i g . 1 where a f l u i d between two plates i s sheared as the top plate moves with v e l o c i t y U i n the x-direction. The v e l o c i t y gradient or shear rate i s given by -y - dv /dy - 7 , and macroscopically i s given by U /£ where 8 i s the plate separation. The stresses generated by the flow act p a r a l l e l to the d i r e c t i o n of shear ( i . e . , shear stresses) and perpendicular to the d i r e c t i o n of shear (normal stresses). The experimentally observable stresses perpendicular to the d i r e c t i o n of flow include the stress a r i s i n g from f l u i d motion and the i s o t r o p i c hydrostatic pressure. I t i s customary to eliminate the i s o t r o p i c pressure by taking the difference between normal stresses, and i t i s i n fact these differences that are experimentally measured: x

yx

x

x

(1)

N

x

- primary normal stress

(2)

difference,

(3)

These stresses are related to the v e l o c i t y gradient, -y , thereby defining the material functions for steady shear flow: yx

r

y x

- -η 7

defines

y x

the viscosity

(4)

2 r

xx

" y y " " * i7yx

yy

"

T

defines the primary normal stress coefficient,

(5)

defines the secondary normal stress coefficient

(6)

2 r

r

z z ™ "*2 ^yx

These material functions generally vary with shear rate. The normal stress c o e f f i c i e n t s are defined i n terms of the square of the v e l o c i t y gradient because the stress difference must be an even power of shear rate; that i s , changing the d i r e c t i o n of the shear (making -y negative) does not change the d i r e c t i o n or sign of the normal stress, whereas changing the d i r e c t i o n of the v e l o c i t y gradient does change the d i r e c t i o n of the shear s t r e s s . yx

Uniaxial Extension/Compression. Consider the flow that either converges or diverges with respect to the ζ-axis as shown i n F i g . 1 . This flow i s produced either by the stretching of a filament (extension) or b i a x i a l stretching of a sheet as i n the case of i n f l a t i n g a balloon (compression i n the z - d i r e c t i o n ) . The measurable stresses are the t e n s i l e normal stresses, and, again, to eliminate i s o t r o p i c pressure terms, the difference i n stresses are used to define the elongational v i s c o s i t y material function:



yy

-η-if

Secondary normal stress difference:

Primary normal stress difference:

Τχν =

Shear stress:

-y = Shear rate, velocity gradient

v„ =

Steady shear

t

v

= 0

3 5

/ω

βν

n ω

*

0

y

M

- Ύο V

Tyv

(f)

-

-*ît)

(12)

where r

cos^ = η'-y max

max

, and r max

s i n ^ s «"γ

.

max


(13)

22

POLYMERS AS R H E O L O G Y MODIFIERS 11

This defines the two dynamic v i s c o s i t y c o e f f i c i e n t s η' and η . At low frequencies η' approaches the zero-shear-rate v i s c o s i t y measured i n steady shear. Alternately, c o e f f i c i e n t s can be defined i n terms of the maximum s t r a i n instead of the s t r a i n rate. r

- -G'7 yx

cos(û>t) - G"7

sin(ot)

max

(14)

max

where r

cos^ = G ' 7


max

, r max

s i n ^ = G"7

max

.

(15)

max

This defines the two functions G' and G" which are the storage and loss moduli, respectively. G', proportional to the stress in-phase with s t r a i n , provides information about the e l a s t i c i t y of a material. For example, an i d e a l e l a s t i c rubber band would have a l l of i t s stress in-phase with s t r a i n or displacement. G", the loss modulus, i s proportional to stress out-of-phase with displacement and, therefore, in-phase with rate-of-displacement or shear-rate. For a purely viscous l i q u i d , a l l of the stress would be out-of-phase with displacement. I t should be kept i n mind that l i n e a r v i s c o e l a s t i c i t y assumes that the stress i s l i n e a r l y proportional to s t r a i n and that the stress response involves only the f i r s t harmonic and not higher harmonics i n frequency ( i . e . , the stress i s a s i n u s o i d a l ) . Experimentally,both of these conditions should be verified. These l i n e a r v i s c o e l a s t i c dynamic moduli are functions of frequency. They have proven to be sensitive probes of the structure of polymer solutions and gels. Figure 2 shows the dynamic moduli for a polymer solution during gelation (7). The material begins as a solution i n F i g . 2a and ends as a s o l i d gel i n F i g . 2d. For a polymer solution at low frequency, e l a s t i c stresses relax and viscous stresses dominate with the r e s u l t that the loss modulus, G", i s higher than the storage modulus, G'. Both decrease with decreasing frequency, but G' decreases more quickly. For a gel the stress cannot relax and, therefore, i s independent of frequency. Also, because the gel i s highly e l a s t i c the storage modulus, G' i s higher than the loss modulus, G". Linear v i s c o e l a s t i c measurements can also be used i n conjunction with c l a s s i c a l polymer k i n e t i c theory to relate the storage modulus of a gel to the number density of c r o s s l i n k s . By following the storage modulus with time, the chemical k i n e t i c s of gel formation can be measured (8,9). Polymer k i n e t i c theory (6) shows that the frequency independent, low frequency l i m i t of the storage modulus for a gel i s given by G' - G° - ι/kT + G

en

(16)

where G° i s the equilibrium shear modulus, ν i s the number density of network strands, k i s Boltzmann's constant, Τ i s the absolute temperature, and G i s a contribution a r i s i n g from entanglements that are not covalent crosslinks. en


2. P R U D ' H O M M E

10

10


3

:

10

10' ο

or/ ο

23


/ ζ *

UJ

10

Ë /

G

'

(A) -12°C

10 1 0.1

0.1 1 10 10 FREQUENCY, RAD/S

2

(C) -18°C 1 10 10 FREQUENCY, RAD/S

2

ο ο

0.1

1 10 10* FREQUENCY, RAD/S

10

2

1 10 10 FREQUENCY, RAD/S

2

Figure 2. Dynamic moduli versus frequency during the process of gelation. The material is polystyrene in carbon disulfide that gels upon cooling. (Reproduced from reference 7. Copyright 1983 American Chemical Society.)


POLYMERS AS R H E O L O G Y MODIFIERS

24 Step Shear Rate

In this experiment, the shear rate i s changed instantaneously from one value to another and the shear stress i s monitored with time. Most commonly, the i n i t i a l shear rate i s zero and the r e s u l t i n g material function i s the "stress growth function":

r

xy

(t) - 5 ( 7

t)7

xyi

(17)

xy

which defines the growth from an i n i t i a l shear rate 7 to the f i n a l shear rate 7 . There are corresponding functions f o r the transient normal stresses: x

y

i


x y

r

xx

- r

yy

- Ν (t) • 1

r yy - r z z - Η2 (t)

t

1

(7

- *2 «

xyi 4

, t)

-y

(18)

2

xy

2

xyi

. t) -γxy

(19)

Step Strain. In step s t r a i n experiments, an instantaneous s t r a i n i s applied to the material and the decay i n the stress i s monitored with time. This defines the shear modulus, G(t), f o r an applied shear s t r a i n of magnitude 7 : 0

r

x y

(t) - G(t) 7

0

defines shear modulus

(20)

and i t defines the Young's modulus, E ( t ) , i f an elongational s t r a i n of magnitude c i s applied. 0

r„(t)

- r ( t ) - E(t) c y y

0

defines Young's modulus

(21)

Constant Stress Creep Experiments. The creep test i s the inverse of the step shear rate experiment; a constant stress r i s applied to the material, and the s t r a i n i s monitored with time. This defines the compliance J : x y

7

xy

(t) - J ( t ) r°

xy

defines the compliance J(t)

(22)

Thixotropic Loop. A deformation h i s t o r y that provides q u a l i t a t i v e information about time-dependent f l u i d rheology i s the thixotropic loop where the shear rate i s continuously ramped from zero to a higher value over a prescribed time period. The r e s u l t i n g shear stress i s measured. This test i s sensitive to the k i n e t i c s of structure evolution which can be important i n aggregated c o l l o i d a l dispersions. I f the structures i n the material are broken apart by shear and cannot reform during the time of the shear rate ramp, then the stresses during the decreasing shear rate ramp w i l l be lower than the stresses during the increasing l e g . The region between the stress and shear rate curves i n the increasing and decreasing ramps i s known as the "thixotropic loop". I t i s d i f f i c u l t to quantify the results observed i n a thixotropic loop experiment because the response i s a complex convolution of the k i n e t i c s of shear induced structure breakdown and the k i n e t i c s of


2.

PRUD'HOMME


25

aggregation. But the thixotropic loop often provides a useful "finger p r i n t " of the material. Experimental Geometries and Simple Flows. Although the material functions are defined for the flows s p e c i f i e d i n the previous section, i t i s often most convenient to measure the material functions using alternate geometries or experiments that approximate the i d e a l flow geometry. Table I gives several examples of geometries from which material functions can be determined for low v i s c o s i t y f l u i d s (10).


Examples of Material Behavior Since rheology i s frequently used as a probe of molecular structure and interactions, i t i s h e l p f u l to have a general idea of what the material functions look l i k e for commonly encountered fluids. Steady Shear V i s c o s i t y and Normal Stresses. Low molecular weight f l u i d s and resins generally are Newtonian f l u i d s . which means that they have a constant v i s c o s i t y independent of shear rate. They also display no e l a s t i c normal stresses. For polymer melts, f i l l e d polymers, polymer solutions and dispersions, the v i s c o s i t y i s not constant, but decreases at higher shear rates as shown i n F i g . 3. Whereas many systems display shear thinning v i s c o s i t i e s , only long chain polymeric molecules exhibit high values of e l a s t i c normal stresses. These e l a s t i c stresses are responsible for phenomena such as die swell, where a polymer extruded through an o r i f i c e swells to a diameter greater than the o r i f i c e diameter. S o l i d f i l l e r s reduce the l e v e l of normal stresses as shown i n F i g . 4, which shows the v i s c o s i t y and normal stress of a polypropylene melt f i l l e d with 50% wt calcium carbonate f i l l e r (11,12,13). Two points should be noted, the v i s c o s i t y increases with f i l l e r concentration, and the primary normal stress difference decreases. For dispersions of s o l i d s i n non-polymeric media, the v i s c o s i t y may also show shear thinning (14), exactly l i k e the polymeric analog; but there w i l l be v i r t u a l l y no normal stresses. For both Newtonian and polymeric continuous phases, v i s c o s i t y increases with increasing volume f r a c t i o n up to a c r i t i c a l volume f r a c t i o n above which the v i s c o s i t y diverges to i n f i n i t y and the material w i l l not flow. This c r i t i c a l volume f r a c t i o n i s about 63% for non-interacting, monodisperse, spheres and decreases markedly when the p a r t i c l e s have long aspect r a t i o s (e.g., chopped glass f i b e r s ) or when the p a r t i c l e s are strongly i n t e r a c t i n g as i s the case when p a r t i c l e sizes are below 1 μπι and surface forces become dominant. I t can be increased to about 80% for a broad d i s t r i b u t i o n of sphere sizes. The e f f e c t of p a r t i c l e aspect r a t i o i s seen i n F i g . 5, which i s the v i s c o s i t y versus shear rate for polyamide 6 melts f i l l e d with glass f i b e r s of d i f f e r e n t aspect r a t i o s , a l l at 30% by weight loading (15). The higher aspect r a t i o f i l l e r s produce higher v i s c o s i t i e s . The e f f e c t of p a r t i c l e surface interactions i s seen for the polystyrene calcium carbonate system i n Fig. 6, where decreasing p a r t i c l e size from 17 μπι to 0.07 μπι at the same volume f r a c t i o n of p a r t i c l e s (φ - 30% by volume) r e s u l t s i n a ten-fold increase i n v i s c o s i t y (16). The e f f e c t of surface


26


Table I. Experimental Geometries for Measuring Fluid Rheology Measured Quantities

Experimental Geometry Row in a tub* (capillary viscometer) - Shearing surface

'Jτ*

shear and Line of path line particle

Q Δ/ R L 5

Volume rate of flow Pressure drop through tube Tube radius Tube length

-y = Shear rate at tube wall R

T


= = = =

= Shear stress at tube wall

r

Torsional flow between a cone and disk

R = Radius of circular plate θ = Angle between cone and plate (usually less than 100 mm) W = Angular velocity of cone Τ = Torque on plate F = Force required to keep tip of cone in contact with circular plate 0

0

Shearing surface

" Une of shear and particle path line Torsional flow between parallel plates

R = Radius of disks H = Separation of disks W = Angular velocity of upper disk Τ - Torque required to rotate upper disk F = Force required to keep separation of two disks constant 0

- Line of shear and particle path line Torsional flow between concentric cylinders (Couette geometry)

- Shearing surface

- Line of shear and particle path line

R,,R =

°f inner and outer cylinders H = Height of cylinders WVW2 = Angular velocities of inner and outer cylinders Τ = Torque on inner cylinder R

a

d

i

i

2



2.

27


PRUD'HOMME

Table I. Continued Material Function Determination Flow in a tube (capillary viscometer) /· ι

Ύ ρ

=

T

R

Π»

d In (Ο/π/ΡΠ

- (TgûAr/? ) TRCITR 3

Torsional flow between a cone and disk η(Ύ) '

Torsional flow between parallel plates

2π/? γ 3

L

d l n

Torsional flow between concentric cylinders

η(Ύ) =

2ir/??/y(IV - IV, j 2

γ = WQ/ΘΟ


^J



28

1 U

1(F

ïir

5

ûr*

ûr

1

ΐ

ίο

ίο

ici

5

5

îô

4

Figure 3. Viscosity versus shear rate for a low-density polyethylene melt at several temperatures. Data at shear rates below 5 χ 10~~ s"" were taken on a rotational viscometer, and viscosities at higher shear rates were taken on a capillary viscometer. (Reproduced with permission from reference 23. Copyright 1971 Hanser.) 2

1

Figure 4. Viscosity, η (open symbols), and first normal stress difference, N . (closed circles), as a function of shear stress for polypropylene melts filled with C a C 0 (50% wt) with and without a titanate coupling agent: (•, • ) pure propylene, (O, · ) with titanate treatment, and ( Δ , A) without titanate treatment. (Reproduced with permission from reference 11. Copyright 1981 Society of Plastics Engineers.) 3


2.

PRUD'HOMME

29


10*

FILLER

d [μχη]

•

GLASS FIBERS

10

25

1

Δ

GLASS FIBERS

13.5

25

1

Ο

GLASS FIBERS

10

10

1

ο

GLASS FIBERS

13.5

7.3

1

—•

GLASS BEADS

15-40

1

1


10'

10

10 - 2

°r

NONE

10

10°

10'

SHEAR RATE γ

10'

1

10°

(S" ) 1

Figure 5. Viscosity versus shear rate for polyamide 6 melts filled with glass fibers of different aspect ratios (a ) and diameters (d) at a constant mass fraction of 30% fibers. (Reproduced with permission from reference 15. Copyright 1984 Steinkopff.) f

1

1

'T

τ " '

«

ι

«

PS/CQC03

φ 10

«0.3

\

6

0.07/im

^^^^v

0.5ftm

17/im

^ ^ ^ • Ν Λ •

1

Y

.

PS

.

(s" ) 1

Figure 6. Viscosity versus shear rate for polystyrene melts with C a C 0 fillers of various particle sizes shown on the figure. The filler loading is constant at 30% vol. (Reproduced with permission from reference 16. Copyright 1983 Wiley.) 3



30

treatments i s also seen i n F i g . 4, f o r the two f i l l e d polypropylene melts at the same solids loading with two d i f f e r e n t surfaces: one i s treated with a titanate surface treatment that minimizes p a r t i c l e surface interactions and the other i s untreated. The treated f i l l e r has a lower v i s c o s i t y since p a r t i c l e aggregation i s reduced. Additional information on f i l l e d melt rheology and p a r t i c l e orientation f o r non-spherical p a r t i c l e s i s found i n a recent review (17). Steady shear v i s c o s i t y measurements are also used f o r polymer molecular weight characterization i n two ways. The measurement of the v i s c o s i t y η, of a d i l u t e polymer solution at a succession of concentrations, C , can be used to determine the i n t r i n s i c v i s c o s i t y , [η]:


p

[η] =

il» c -o p

V-P*C f; p

(23)

s

where η i s the solvent v i s c o s i t y . The i n t r i n s i c v i s c o s i t y i s r e l a t e d to molecular weight, M^, through the Mark-Houwink expression: Λ

[if] - Κ lC (24) where Κ and a are constants tabulated f o r each polymer i n standard references (18). Also, the zero shear v i s c o s i t y , η , of a polymer solution or melt can be used to determine i t s molecular weight. Figure 7 (6) shows s i m i l a r v i s c o s i t y molecular weight behavior f o r several amorphous, l i n e a r polymers. The data can be represented by 0

η

0

- M^)

for if < M w

3

η

0

- Κ (Μ^) · 2

4

c

for J* < M w

c

(25) (26)

The constants Kj and K are tabulated (18) and the t r a n s i t i o n from the f i r s t order to 3.4 order dependence on molecular weight comes from the t r a n s i t i o n from unentangled behavior at low molecular weight [corresponding to material II i n the following section] to entangled behavior at high molecular weight [corresponding to material I I I i n the following section]. 2

Uniaxial Extension/Compression. Polymeric melts and v i s c o e l a s t i c f l u i d s display elongational v i s c o s i t i e s as a function of elongational s t r a i n rate as shown i n F i g . 8. The data are shown f o r a polystyrene melt at 170°C at several constant elongation rates (24). The elongational v i s c o s i t y increases with time i n i t i a l l y , may reach a constant asymptotic value that i s equal to three times the zero shear v i s c o s i t y , and then may increase u n t i l the f i b e r breaks. The onset of the r i s e i n v i s c o s i t y i s related to the rate of elongation, and the onset occurs e a r l i e r f o r higher elongation rates. This f i n a l " s t r a i n hardening" i s one of the mechanisms s t a b i l i z i n g the stretching of polymer f i b e r s and i s therefore very desirable. However, i t makes r e l i a b l e "steady elongational v i s c o s i t y " data very d i f f i c u l t to obtain. Most data on f i b e r



2.

PRUD'HOMME


31

Ο Polydimethylsiloxane

i CO C Ο

Polyisobutylene

U Polyethylene Polybutadiene Polytetra-methyl-p-silphenylene, siloxane

Polymethylmethacrylate

. Polyethylene glycol Polyvinyl acetate Polystyrene

Constant + log (cM) Figure 7. Log of the zero-shear-rate viscosity versus log of concentration times molecular weight. The data are shifted along the vertical and horizontal axes; the two shift factors are tabulated for many polymer systems. The molecular weight at the break point between the region of slope 1 and the region of slope 3.4 at higher molecular weight defines the critical molecular weight for entanglement, Mc. (Reproduced with permission from reference 24. Copyright 1968 Springer.)



32


10

_

7

3*

10

0

Polymers as Rheology Modifiers - American Chemical Society

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