Ordered Fluids and Liquid Crystals

22 Analysis and Apparatus for Surface Rheological Measurements

Downloaded by UNIV OF CINCINNATI on May 31, 2016 | http://pubs.acs.org Publication Date: January 1, 1967 | doi: 10.1021/ba-1967-0063.ch022

R. A. BURTON and R. J. MANNHEIMER Southwest Research Institute, San Antonio, Tex.

The mathematical analysis presented here applies to the steady flow of a Newtonian fluid in an infinitely long channel having fixed side walls, a floor moving steadily in a direction parallel to the walls, and a flat, free surface which may exhibit the prop erties of surface viscosity and surface shear rigidity. In the cor responding experimental apparatus the infinitely long channel is simulated by an annulus. Using liquids with different viscosities as test fluids, the general predictions of the analysis are verified for the flow of simple fluids. The proposed ap paratus configuration is adaptable to both insoluble and soluble surface films. This, coupled with the amenability of the config uration to direct analysis, gives promise of a potential for broad application.

Τ η 1869, P l a t e a u (7) i n f e r r e d t h e existence of surface v i s c o s i t y b y o b s e r v i n g t h e d a m p i n g of t h e m o t i o n of a needle l y i n g o n t h e surface of a l i q u i d . S i n c e t h e n n u m e r o u s e x p e r i m e n t a l approaches h a v e been developed to p e r m i t m o r e precise measurement of t h i s a n d o t h e r surface-rheological p r o p e r t i e s ; a n i n c r e a s i n g l y coherent b o d y of d a t a has been a c c u m u l a t e d i n recent years. S u c h measurements are of b r o a d interest since t h e y shed l i g h t o n t h e i n t e r m o l e c u l a r forces at w o r k i n films a n d answer questions r e g a r d i n g t h e p r a c t i c a l p r o b l e m s of f o a m f o r m a t i o n , e m u l s i o n s t a b i l i t y , a n d wave damping. I n effect, surface v i s c o s i t y is d i r e c t l y e q u i v a l e n t t o a significant increase of b u l k v i s c o s i t y i n t h e v i c i n i t y of a l i q u i d - g a s or l i q u i d - l i q u i d interface. T h i s region of enhanced v i s c o s i t y m a y correspond t o a n i n s o l u b l e m o n o l a y e r f l o a t i n g o n t h e surface of a substrate l i q u i d . I t m a y also be p r o d u c e d b y soluble components i n t h e b u l k l i q u i d , w h i c h t e n d t o concentrate at a n o p e n surface or a n interface. 315 Porter and Johnson; Ordered Fluids and Liquid Crystals Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

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T h e r e l a t i o n between surface v i s c o s i t y a n d b u l k v i s c o s i t y is g i v e n b y μ

5

= δ (μα) a

where ( μ ) is t h e e q u i v a l e n t b u l k v i s c o s i t y of t h e surface phase, a n d δ is t h e t h i c k n e s s of t h e surface region of enhanced v i s c o s i t y . μ is called t h e surface v i s c o s i t y a n d represents t h e c o m b i n e d effects of b o t h v i s c o s i t y a n d t h i c k n e s s of t h e surface l a y e r . T h e u n i t s of μ are surface poises or d y n e 6

β

β


8

seconds per centimeter r a t h e r t h a n dyne-seconds per square c e n t i m e t e r of conventional bulk viscosity. T h e t a s k of m e a s u r i n g surface v i s c o s i t y is t w o f o l d . F i r s t , g e o m e t r y m u s t be devised w h e r e i n some easily m e a s u r e d p a r a m e t e r of t h e flow is s t r o n g l y influenced b y t h e surface region of enhanced v i s c o s i t y . S e c o n d , a m a t h e m a t i c a l analysis m u s t be d e r i v e d t o relate t h e measurement o b s e r v a t i o n t o t h e desired q u a n t i t y . W i t h a l , t h e status of measurement t e c h niques a n d a p p a r a t u s has been s u c h t h a t t h e o n l y a r r a n g e m e n t s u i t e d for absolute p r o p e r t y measurements is l i m i t e d t o insoluble surface films w h i l e t h e m o r e b r o a d l y a p p l i c a b l e arrangements h a v e n o t successfully y i e l d e d t o a n a l y t i c a l t r e a t m e n t . T h e present w o r k represents a n a p p r o a c h t o w a r d b r i d g i n g t h i s gap. A m o d i f i e d c o n f i g u r a t i o n is i n t r o d u c e d for a c h a n n e l t y p e viscometer, offering b o t h b r o a d e x p e r i m e n t a l a p p l i c a b i l i t y a n d a m e n a b i l i t y t o m a t h e m a t i c a l analysis. T h i s c o n f i g u r a t i o n is closely enough r e l a t e d t o some of t h e earlier arrangements t o p e r m i t i t s use w i t h e x i s t i n g l a b o r a t o r y procedures, a n d i t is a p p l i c a b l e t o b o t h i n s o l u b l e a n d soluble films.

Figure 1.

Sectional view of channel

T h e p h y s i c a l m e a n i n g of surface v i s c o s i t y m a y best be i l l u s t r a t e d i n t e r m s of t h e proposed measurement t e c h n i q u e itself. C o n s i d e r a n a n n u l a r c h a n n e l w i t h s t a t i o n a r y w a l l s a n d v e r t i c a l axis of s y m m e t r y , t h a t is p a r t i a l l y filled w i t h a l i q u i d t o a u n i f o r m d e p t h , as i l l u s t r a t e d i n cross-section i n F i g u r e 1. T h e floor of t h e c h a n n e l is capable of t a n g e n t i a l s l i d i n g . When t h e floor is set i n u n i f o r m m o t i o n , i t gives rise to a corresponding m o t i o n i n

Porter and Johnson; Ordered Fluids and Liquid Crystals Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

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317

t h e l i q u i d , where t h e free-surface speed w i l l be slower t h a n t h e floor speed as a consequence of t h e influence of t h e c h a n n e l w a l l s . T h i s speed c a n be observed i n t e r m s of the m o t i o n of particles of n o n w e t t i n g m a t e r i a l d u s t e d o n t o t h e l i q u i d surface. I f a film t h a t shows surface v i s c o s i t y is a p p l i e d to t h e l i q u i d , t h e s u r face speed w i l l be f u r t h e r reduced. I f t h e film shows surface r i g i d i t y (ef f e c t i v e l y , i n f i n i t e surface v i s c o s i t y ) , t h e surface m o t i o n w i l l be b r o u g h t to a complete h a l t , a n d t h e net effect of t h e m o v i n g floor w i l l be to a p p l y stress r a t h e r t h a n m o t i o n to the film. O f t e n a r i g i d " film w i l l b r e a k u p at some l i m i t i n g stress w h i c h c a n be t e r m e d surface y i e l d s t r e n g t h . F o l l o w i n g s u c h a b r e a k u p , t h e film m a y e x h i b i t surface v i s c o s i t y i n v a r y i n g degrees. Downloaded by UNIV OF CINCINNATI on May 31, 2016 | http://pubs.acs.org Publication Date: January 1, 1967 | doi: 10.1021/ba-1967-0063.ch022

44

W i t h q u a n t i t a t i v e relationships a m o n g surface v e l o c i t y , t h e geometric parameters, t h e s u b s t r a t e fluid v i s c o s i t y , a n d t h e floor speed, t h i s a r r a n g e m e n t n o t o n l y demonstrates surface v i s c o s i t y effects b u t also p r o v i d e s a means for its measurement. Prior

Developments

T h e reasons b e h i n d t h e specific choice of a p p a r a t u s geometry c a n best be s h o w n b y a brief review of p r i o r w o r k . T h e earliest " c a n a l t y p e " s u r face v i s c o m e t e r was i n t r o d u c e d b y D e r v i c i a n a n d J o l y (3). I n this ap p a r a t u s , a n insoluble m o n o l a y e r is floated o n a s u b s t r a t e fluid i n a s t r a i g h t c h a n n e l . T h e f i l m is forced t o flow t h r o u g h t h e c h a n n e l b y m o v e m e n t of a floating b a r r i e r . T h i s m o t i o n is resisted p r i n c i p a l l y b y surface v i s c o s i t y . T h u s , t h e s m a l l force r e q u i r e d to p r o p e l t h e film at a g i v e n speed m a y be measured a n d used t o d e t e r m i n e t h e surface v i s c o s i t y of t h e film. A r e l a t i v e l y complete t h e o r e t i c a l t r e a t m e n t has been p r o v i d e d b y H a r k i n s a n d K i r k w o o d (5) for insoluble films w i t h N e w t o n i a n surface v i s c o s i t y i n deep channels. A c t u a l measurements are t y p i c a l l y m a d e i n s h a l l o w channels, however, w h i c h are f o r m e d b y floating t h e c h a n n e l b o u n d a r i e s o n t h e l i q u i d surface. T h i s m e t h o d is n o t a p p l i c a b l e to soluble surface films, w h i c h t e n d to diffuse t h r o u g h t h e substrate fluid a n d pass b e h i n d t h e b a r r i e r . N e v e r theless, t h e most accurate values of surface v i s c o s i t y a v a i l a b l e h a v e been produced by this approach. E w e r s a n d S a c k (4) h a v e developed a related a p p a r a t u s consisting of a v e r t i c a l w a l l e d c h a n n e l between t w o reservoirs, t h r o u g h w h i c h t h e s u b strate l i q u i d m o v e s as a consequence of a h e a d difference between t h e reservoirs. S m a l l n o n w e t t a b l e particles floated o n t h e surface i n d i c a t e t h e speed of surface elements of t h e fluid a l o n g t h e center l i n e of t h e c h a n n e l . T h i s m o t i o n has been related to b u l k - f l u i d m o v e m e n t b y means of a t h o r o u g h m a t h e m a t i c a l t r e a t m e n t , w h i c h is a p p l i c a b l e to N e w t o n i a n surface films. T h u s , t h e force measurement i n t h e previous scheme is replaced here b y a surface v e l o c i t y a n d b u l k - f l o w measurement. T h e o n l y serious d i f f i c u l t y w i t h t h i s a r r a n g e m e n t is t h a t t h e t r a n s p o r t e d surface m a t e r i a l tends to a c c u m u l a t e i n t h e lower reservoir. T h u s , a surface pressure g r a -



318

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d i e n t is created w h i c h affects t h e surface flow a n d l i m i t s t h e p r a c t i c a l use fulness of the a p p a r a t u s . D a v i e s (2) has i m p r o v e d o n t h i s design w i t h h i s c i r c u l a r " v i s c o u s t r a c t i o n " i n s t r u m e n t , w h i c h consists of a n a n n u l a r brass c h a n n e l , f o r m e d b y t w o concentric knife-edge r i n g s . T h e s e are coated w i t h w a x a n d t h e n lowered u n t i l t h e y m a k e contact w i t h t h e surface of a l i q u i d i n a P e t r i d i s h . A s t h e d i s h is r o t a t e d about its axis, t h e b u l k m o t i o n of t h e f l u i d tends to p r o v i d e a t r a c t i o n o n t h e l i q u i d surface between t h e t w o rings. A s before, t h e speed of a suspended p a r t i c l e o n t h e surface provides a n i n d i c a t i o n of t h e surface v i s c o s i t y . A l t h o u g h D a v i e s was successful i n e l i m i n a t i n g t h e surface pressure effects t h a t h a d p l a g u e d t h e p r e v i o u s c h a n n e l viscometers, he has conceded t h a t " t h e m a t h e m a t i c a l t r e a t m e n t of t h e r e t a r d a t i o n of t h e surface r e l a t i v e t o t h e b u l k of t h e l i q u i d is r a t h e r c o m p l i c a t e d " (2). D a v i e s , therefore, h a d t o depend u p o n c a l i b r a t i n g his s y s t e m w i t h i n s o l u b l e films. T h e surface viscosities of these films h a d p r e v i o u s l y been d e t e r m i n e d b y t h e c a n a l m e t h o d of D e r v i c i a n a n d J o l y . T h i s a p p r o a c h has m a d e i t difficult t o a p p l y t h e a p p a r a t u s to a b r o a d v a r i e t y of e x p e r i m e n t a l c o n d i tions. F u r t h e r m o r e , t h e c a l i b r a t i o n d a t a h a v e been l i m i t e d to i n s o l u b l e films o n aqueous s u b s t r a t e s ; therefore, substrate l i q u i d s w i t h b u l k v i s c o s i ties a p p r e c i a b l y different f r o m w a t e r c o u l d n o t be i n v e s t i g a t e d b y t h i s method. T h e a p p a r a t u s c o n f i g u r a t i o n proposed i n t h i s p a p e r retains t h e a n n u l a r flow features of D a v i e s ' a p p a r a t u s w h i l e e l i m i n a t i n g t h e h a r d - t o - a n a l y z e r i n g configuration. Since i t c a n be dealt w i t h i n t e r m s of flow i n a u n i f o r m c h a n n e l , i t m a y also be considered t o r e t a i n t h e most f a v o r a b l e feature of t h e a p p a r a t u s of E w e r s a n d S a c k w i t h o u t suffering f r o m t h e p r o b l e m of surface-pressure b u i l d u p . I n a d d i t i o n t o those c h a n n e l v i s c o m e t e r s discussed here there is a sep arate t y p e k n o w n as t o r s i o n a l viscometers, where surface v i s c o s i t y is m e a s u r e d i n t e r m s of t h e t r a c t i o n o n a w i r e , t r a v e r s e d lengthwise o n t h e surface of a l i q u i d . A t t h i s w r i t i n g , none of these is k n o w n t o h a v e been s u b j e c t e d t o a complete a n a l y s i s . J o l y (6) says, " f u r t h e r m o r e , c a l c u l a t i o n s based o n the correct e q u a t i o n w o u l d d e p e n d u p o n t h e exact shape of the a p p a r a t u s . A s these calculations h a v e so far n o t been c a r r i e d out i n a n y case, t h e v a l u e s o b t a i n e d b y t h e r o t a t i o n m e t h o d are i n c o r r e c t a n d c e r t a i n l y too h i g h . " Nevertheless, s u c h approaches m a y offer special advantages a n d s h o u l d be i n v e s t i g a t e d f u l l y . T h e y m u s t be considered as separate topics, h o w e v e r f r o m the c h a n n e l viscometers discussed here.

Theoretical

Background

I n a n idealized v e r s i o n of t h e proposed c o n f i g u r a t i o n ( F i g u r e 1), a n o r i g i n for C a r t e s i a n coordinates is p l a c e d at one w a l l a n d at t h e floor of t h e


22.

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319

T h e ?/-axis is d i r e c t e d across t h e c h a n n e l , t h e χ axis is d i r e c t e d

u p w a r d f r o m t h e c h a n n e l floor, a n d t h e 2-axis is i n a d i r e c t i o n n o r m a l to t h e (x, y) plane.

T h e f o l l o w i n g a n a l y s i s is m a d e u n d e r the a s s u m p t i o n s t h a t

t h e free fluid-surface is flat, t h a t t h e a n n u l a r space between t h e concentric rings c a n be t r e a t e d as a s t r a i g h t a n d i n f i n i t e l y l o n g c h a n n e l , t h a t t h e flow i n t h e c h a n n e l is unaccelerated a n d l a m i n a r , a n d t h a t b o t h surface a n d b u l k viscosities are N e w t o n i a n .

T h e o n l y v e l o c i t y c o m p o n e n t w i l l be w, w h i c h

is d i r e c t e d a l o n g t h e 2-axis; t h e m a g n i t u d e of w w i l l be a s i n g l e - v a l u e d scalar f u n c t i o n of χ a n d y; of

flow.

a n d there w i l l be no pressure gradients i n t h e d i r e c t i o n

U n d e r these conditions, t h e N a v i e r - S t o k e s e q u a t i o n reduces t o


L a p l a c e ' s e q u a t i o n i n t w o dimensions for t h e substrate fluid, where μ is constant t h r o u g h o u t :

T h e b o u n d a r y c o n d i t i o n s at t h e w a l l s are s u c h t h a t w = 0 at y = 0 w = 0 at y = yο

(2)

A t t h e floor, where w is constant, w i t h m a g n i t u d e w , t h e b o u n d a r y c o n d i f

t i o n is s u c h t h a t w = w at χ = 0

(3)

f

A t t h e u p p e r surface of t h e b u l k or substrate flow t h e b o u n d a r y c o n d i t i o n is set b y t h e effects of surface v i s c o s i t y .

E w e r s a n d S a c k (4) h a v e s h o w n

t h i s t o be specified b y E q u a t i o n 4. d_w 1X3

dw

(4)

dy*

Tj dw\ H e r e μ, ^ is recognized as t h e l o c a l shear stress, r , a c t i n g o n a d i f f e r e n t i a l surface element. T h e t e r m at t h e left results w h e n i t is assumed t h a t xz

s

t h e surface film is t h i n enough t o share w i t h t h e s u b s t r a t e t h e v e l o c i t y w h i c h is a f u n c t i o n of y o n l y .

w, s

T h e gradient of t h i s v e l o c i t y i n t h e p l a n e of

t h e surface gives rise to a l i n e stress i n d i r e c t a n a l o g y to t h e viscous r e sistance of b u l k

flow.

μ is t h e v i s c o s i t y of t h e substrate flow a n d is as 6

s u m e d t o h a v e a constant m a g n i t u d e t h r o u g h o u t t h e b u l k

fluid.

μ is s u r 8

face v i s c o s i t y a n d for the present analysis is assumed to h a v e a c o n s t a n t m a g n i t u d e at a n y p o i n t i n t h e surface

film.


320

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FLUIDS A N DLIQUID

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Equations

A general s o l u t i o n f o r E q u a t i o n 1, t h a t satisfies t h e b o u n d a r y c o n d i t i o n s of E q u a t i o n 2, m a y be w r i t t e n as (a e

Σ

w

s i n mry/y

+ b e-"" )

nrx/yo

n

/vo

n

(5)

0

w h e r e η = 1, 2, 3, . . . . T h e p r o b l e m r e m a i n i n g is t o a p p l y t h e l o w e r a n d u p p e r surface b o u n d a r y c o n d i t i o n — i . e . , a t χ = 0 a n d χ = x —to

e v a l u a t e coefficients a a n d b .

0

n

n


I t i s h e l p f u l i n t h i s t a s k t o i n t r o d u c e t h e dimensionless coordinates : =

x

πχ/ijo

Y

=

(6)

ν ο

ml

from w h i c h E q u a t i o n 5 becomes: w=JT

(«ηβ

ηΧ

s i n nY

+ b e~ ) n

nX

(7)

n=l

A t t h e c h a n n e l floor, X = 0 a n d w = w , so t h a t E q u a t i o n 7 b e c o m e s : f

(«η + b„) s i n nY,

w = Σ f

0 < Υ < π

(8)

n=l

T h i s is recognized t o be i n t h e f o r m of a F o u r i e r sine series o v e r t h e i n t e r v a l 0 < Υ < π.

D r a w i n g upon the Dirichlet integral technique to evaluate

t h e coefficient (a + b ) n

(β

η

n

+ b) = — n

I J

7Γ

w s i n η YdY =

(n, o d d )

f

0

(η, even)

(«η + Κ) = 0

(9)

T h e r e q u i r e m e n t of s y m m e t r y about t h e c h a n n e l center l i n e c a n be satisfied o n l y b y o d d v a l u e s of n.

F r o m t h i s , a m a y be d e r i v e d a n d i n s e r t e d i n t o n

E q u a t i o n 7, t o o b t a i n :

W

=Σ n=l

( M _ ) »* bn

L \

e

/

^7T

+

6ηβ

-**1

s i nn

F

( n jo d d )

( 1 0 )

J

T o e v a l u a t e b , t h e b o u n d a r y c o n d i t i o n s a t t h e fluid surface m a y b e n

used, as g i v e n i n E q u a t i o n 4, w h i c h , i n t h e t r a n s f o r m e d coordinates, b e comes :

μ.

ο w y»-

dw

(11)

' ax

d

where ν = 2/ μ&/ττ a n d D = ^ / z / 0

0

0


(12)

22.

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M A N N H E i M E R

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321

Measurements

F r o m E q u a t i o n s 10 a n d 1 1 : -

- μ , Σ Ρ e η=ι L ηπ

nvD

Z \4wf7Γ

n=l

2b s i n h nwD n

2b

n

cosh ηπΡ "

l-W

\

J

s i n nY

odd)

(η,

, ,, (n, o d d )

v

λ

sin η Γ

(13)

J

?1

A g a i n , e q u a t i n g coefficients of corresponding t e r m s , t h i s is reduced t o _

fe^ ηπ

M s

L

_

2 h n

s

i

n

n

h

r

D

\

,

=

fe » LnV

_

r

J

2b oshn,Dl η J nC

E q u a t i o n 14 is s o l v e d for b t o y i e l d :


n

b n

(W? + l)4w e * 2ηπ[(ημ /ν) s i n h ηπΌ + cosh ηπΌ] f

=

n

,

D

o

d

d

)

( 1 5 )

'

8

T h i s result c a n be used i n E q u a t i o n 10 to give t h e fluid speed, w, at a n y point i n the channel.

O f p a r t i c u l a r interest is the surface speed, w ,

for

s

w h i c h E q u a t i o n 10 becomes: w

8

2 6 s i n h ηπΌ

= X)

s i n nY

n

n=i L

(n, odd)

(16)

J

ηπ

E l i m i n a t i n g 6 b e t w e e n E q u a t i o n 15 a n d E q u a t i o n 16 y i e l d s : n

4 w / s i n n F = — Z^TT / \ · u FT , r ™ π ^ ί η [ ( η μ / ϊ ' ) s i n h ηπΖ) + cosh rnrD] A l o n g t h e center l i n e of t h e surface, Υ = π/2 a n d w = w

1

s

, (n, odd)

-

7

β

8

w

sinn7r/2

4u)f ^

c

t

/10

= — - 2^ "77 / \ - u FTI û 7vi (> ) π " ^ L Î ^ M s A ) s m h ?i7rZ) + cosh n7ri>J F o r deep c h a n n e l s — i . e . , Ό > 2/π—the t e r m s i n t h i s series for w > w

n

c

negligible.

. (17)

n f 7

o d d

v

( ) 1 8

1 are

T h u s E q u a t i o n 18 m a y be s i m p l i f i e d to w

°

-

u

ι \ · ^Τ\Λ_

ΪΓ-ΪΫι

> —

D

(

1 9

)

7r[ (μ /ν) s i n h πΏ + cosh πΌ\ π F o r example, at D = 2/ττ, t h e error i n c u r r e d i n u s i n g E q u a t i o n 19 i n s t e a d &

of E q u a t i o n 18 is less t h a n 0 . 5 % .

S i n c e t h e e x p e r i m e n t a l error w i l l n o r

m a l l y be greater t h a n t h i s , t h e use of E q u a t i o n 19 c a n be j u s t i f i e d i n m o s t instances.

S t a t e d d i f f e r e n t l y , use of a deep c h a n n e l is r e c o m m e n d e d for

e x p e r i m e n t a l measurements e q u a t i o n of

since i t p e r m i t s great s i m p l i f i c a t i o n of

the

flow.

I f there is no film c o v e r i n g t h e surface, t h e n μ

8

= 0, a n d t h e center

l i n e surface speed i n a deep c h a n n e l reduces t o :

v

' * ï £ f c D

where t h e asterisk denotes t h a t μ

8

=

D

>

2

/

*

0.


( 2 0 )

322

ORDERED

FLUIDS

AND

LIQUID

CRYSTALS

F u r t h e r m o r e , b y t a k i n g t h e differences of E q u a t i o n s 19 a n d 20, t h e f o l l o w i n g relationships are o b t a i n e d : ^ ^ = s = w

W t a n h x D \v I

c

B u t since t a n h TD = 1 for D >

*

-

w

Z > > -

(21)

π

2/π

- ± » i

w

D

>

2 L

(

22)

W

c

T h i s e q u a t i o n m a y be a p p l i e d d i r e c t l y to c o m p u t e surface f r o m e x p e r i m e n t a l measurements of w a n d Downloaded by UNIV OF CINCINNATI on May 31, 2016 | http://pubs.acs.org Publication Date: January 1, 1967 | doi: 10.1021/ba-1967-0063.ch022

c

Derivation

of Stress Equation

T h e q u a n t i t y μ 1^

viscosity

w *.

for a Rigid

c

Film

m a y be dealt w i t h as a line stress a c t i n g i n t h e s u r

8

face a n d g i v e n t h e s y m b o l T .

I t w i l l be i n u n i t s of dynes per c m . a n d

S

w i l l bear direct analogy to surface tension, except t h a t i t is a t a n g e n t i a l r a t h e r t h a n n o r m a l stress. R e t u r n i n g to E q u a t i o n 4 a n d i n s e r t i n g dr, _ dy Or, i n the transformed

dw μ ύ

(23)

dx

coordinates dr, _

dw

dY ~

βΧ

μι

(24) X=TTD

F r o m previous r e l a t i o n s ; — i . e . , E q u a t i o n s 10 a n d 1 4 , — i t c a n be s h o w n t h a t dw I dX I

=

s i n nY

-4wf

X=TTD

riri [(ν/ημ )

7Γ

8

,

cosh mrD + s i n h mrD]

_

( 2 g )

'

F u r t h e r m o r e , since t h e stress is zero at m i d - c h a n n e l a n d a m a x i m u m at either channel w a l l (i.e., Y = 0 or Υ = π), i n t e g r a t i n g E q u a t i o n 24 y i e l d s :

π or for D >

t=i L w μ ) cosh nwD + n s i n h mrD\ 8

§ 7Γ

k . l m a x = -η ^ /, μ . ). cosh Γ μirD η ^+ s-i nκh TD] nï 4

X

(

D

> ~ π

( ) 27

F o r a r i g i d f i l m , μ —> c o , a n d ν/μ —> 0 ; therefore, t h e y i e l d v a l u e is 8

8

4

π s i n h irD

=

D > — π


(28) v

22.

BURTON

AND

M A N N H E i M E R

Surface

Rheological

Measurements

323

T h u s , t h e y i e l d v a l u e c a n be o b t a i n e d b y m e a s u r i n g the m i n i m u m floor speed, w , r e q u i r e d t o i n i t i a t e surface flow i n a specified c h a n n e l geometry. f

F o r systems i n w h i c h t h e c h a n n e l does not correspond t o t h e exact t h e o r e t i c a l m o d e l (discussed below), c a l i b r a t i o n of t h e a p p a r a t u s w i t h a s i m p l e l i q u i d w i l l p r o v i d e t h e r e q u i r e d v a l u e of


Experimental

Verification

of Principal

w*. Assumptions

Questions m a y be raised as t o t h e consequences of t h e basic a s s u m p t i o n s used i n t h e d e r i v a t i o n s since these neglect b o t h t h e presence of a meniscus a n d t h e c u r v a t u r e of t h e c h a n n e l . T o answer these a n d p r o v i d e e x p e r i m e n t a l s u p p o r t for the d e r i v a t i o n s , a n a p p a r a t u s was c o n s t r u c t e d as s h o w n i n F i g u r e 2 a n d t h e d i m e n s i o n e d , s c h e m a t i c cross-section ( F i g u r e 3).

Figure 2.

Deep-ckannel surface-viscosity apparatus

T h e c h a n n e l is f o r m e d b y t w o stainless steel c y l i n d e r s , m o u n t e d o n a t r a n s p a r e n t p l a s t i c s u p p o r t . A m i c r o m e t e r p r o v i d e s a means t o measure t h e d e p t h of t h e m i d - c h a n n e l surface of t h e l i q u i d , a n d t h e entire c h a n n e l c a n be raised a n d lowered or leveled r e l a t i v e to t h e floor of t h e l i q u i d - c o n t a i n i n g d i s h . U s i n g t h i s arrangement, several t y p e s of tests were c a r r i e d o u t to d e m o n s t r a t e different aspects of its o p e r a t i o n . I n each case, t h e rings were a d j u s t e d a n d leveled, t h e l i q u i d was t i t r a t e d i n t o t h e d i s h t o p r o d u c e a preselected d e p t h , t h e t u r n t a b l e w a s set i n m o t i o n , a n d t h e progress of a floating T e f l o n p a r t i c l e was observed as i t m o v e d a l o n g i n t h e l i q u i d surface. B e c a u s e of t h e c o n c a v e - u p w a r d meniscus, t h e p a r t i c l e t e n d e d t o r e m a i n at m i d - c h a n n e l a n d t h u s gave a measure of t h e center l i n e surface v e l o c i t y , w . T h e measurement t a k e n was the particle time, the t i m e r e q u i r e d for one o r b i t of t h e p a r t i c l e a r o u n d t h e c h a n n e l . c



324

ORDERED

FLUIDS AND

LIQUID

CRYSTALS

53 mm.

Figure 3.

Table I.

Schematic cross-section of deep-channel surface-viscosity apparatus

Typical Measurements** of Particle Time, (t *),for Pure Liquids c

Material

Surface Tension, Dynes/Cm.

%Cetane

25

Mineral oil

30

Distilled water

73

Bulk Viscosity, Cp.

t * Seconds

150

c

149 149 150 149 149 150 149 152 153

° Representative of more than 100 determinations.

O n e q u e s t i o n t o be answered w o u l d concern t h e b e h a v i o r of fluids of g r e a t l y d i f f e r i n g v i s c o s i t y a n d surface t e n s i o n , w h i c h o n t h e basis of o t h e r w o r k w o u l d n o t be expected t o e x h i b i t surface v i s c o s i t y . T a b l e I shows t h e results for a p a r t i c u l a r set of runs for w a t e r , cetane, a n d a w h i t e m i n e r a l o i l w i t h o u t a d d i t i v e s . A l t h o u g h t h e surface t e n s i o n r a n g e d f r o m 25 t o 73 dynes per c m . a n d t h e b u l k v i s c o s i t y r a n g e d f r o m 1 t o 150 cp., t h e p a r t i c l e t i m e , t *, r e m a i n e d t h e same for each. Reference t o E q u a t i o n 20 for w*, indicates t h a t t h i s i n v a r i a n c e s h o u l d be expected since n e i t h e r v i s c o s i t y n o r surface t e n s i o n appears i n t h e d e r i v e d e q u a t i o n for p a r t i c l e v e l o c i t y , w*. A l s o these d a t a show t h a t despite hypotheses of a surface s t r u c t u r e for w a t e r , t h i s fluid showed no significant d e p a r t u r e f r o m t h e b e h a v i o r of t h e cetane, w h i c h is n o n p o l a r a n d w o u l d be expected t o h a v e no s u c h s t r u c t u r e . T h o u g h no f u r t h e r reference is m a d e t o fluid t y p e because of t h e observed c


22.

B U R T O N

AND

M A N N H E I M E R

Surface

Rheological

Measurements

325

s i m i l a r i t i e s of b e h a v i o r , t h e d a t a i n the r e m a i n i n g comparisons were also checked for m o r e t h a n one fluid. E v e n t h o u g h n o surface-tension effect was a p p a r e n t i n T a b l e I , r u n s w i t h different c h a n n e l w i d t h s i n d i c a t e t h a t t h e shape of t h e meniscus mayh a v e a measurable, t h o u g h s m a l l , influence o n t h e m e a s u r e d surface speed. F o r example, i n F i g u r e 4 a c o m p a r i s o n is s h o w n f o r o p e r a t i o n w i t h t h e l o w e r edges of t h e c h a n n e l i n p h y s i c a l contact w i t h t h e m o v i n g floor. D a t a


for t w o c h a n n e l w i d t h s are c o m p a r e d w i t h t h e p r e d i c t i o n s of E q u a t i o n 18. T o n o n d i m e n s i o n a l i z e t h e p l o t , p a r t i c l e speed w* is expressed i n r a t i o t o

i

1.0

:

" .05

0

Ο y = 1.12 cm. 0

Δ y 0.55cm. 0

s

0.2 0.4 0.6 0.8 1.0 DEPTH TO WIDTH RATIO, x /y 0

0

Figure 4» Effect of channel width on agreement between ex perimental and theoretical surface velocity measurements for fluids without surface viscosity

floor speed, a n d fluid d e p t h is expressed as a f r a c t i o n of c h a n n e l w i d t h . T h o u g h close agreement w i t h t h e t h e o r e t i c a l p r e d i c t i o n is i n d i c a t e d , i t is clear t h a t t h e w i d e r c h a n n e l gave t h e b e t t e r results. A l t h o u g h y e t u n k n o w n factors m a y u l t i m a t e l y account for t h i s difference, t h e f a v o r e d e x p l a n a t i o n is t h a t t h e meniscus i n t h e w i d e c h a n n e l is t h e m o r e n e a r l y flat a n d t h u s i n closer agreement w i t h t h e assumed geometry. Attempts to p r o d u c e a flat meniscus t h r o u g h c o n t r o l of c o n t a c t angle a t t h e w a l l h a v e p r o v e d difficult a n d indecisive. E x p e r i m e n t s w i t h s u c h a flat meniscus h a v e p r o v e d fruitless since t h e p a r t i c l e t e n d e d t o m i g r a t e t o t h e c h a n n e l w a l l s . I n a n y event, t h e effect appears t o be s m a l l i n i t s t o t a l influence o n w . I t w o u l d be expected t o be e v e n s m a l l e r i n its influence o n t h e c a l c u l a t i o n of surface v i s c o s i t y f r o m measurements since t h e p e r t i n e n t velocities, w a n d w*j appear i n r a t i o t o one a n o t h e r a n d b o t h w o u l d b e expected t o be affected s i m i l a r l y b y t h e meniscus. c

c


326

ORDERED

FLUIDS A N D

LIQUID

CRYSTALS

F i g u r e 5 shows t h e effect of clearance between t h e lower l i p of t h e c h a n n e l a n d t h e m o v i n g floor. T h e s e measurements d e m o n s t r a t e t h a t t h e general effect of clearance is t o render t h e a p p a r e n t d e p t h - w i d t h r a t i o greater t h a n t h e a c t u a l m a g n i t u d e . T h e s e observations h a v e t w o i m p l i c a t i o n s : t h a t a s m a l l gap at t h e contact between c h a n n e l a n d floor w i l l n o t p r o d u c e a d i s p r o p o r t i o n a t e effect o n t h e readings, a n d t h a t t h e a p p a r e n t flow b e h a v i o r w i t h a large gap is v e r y m u c h l i k e t h a t w i t h o u t a gap.


ι

1

1

1

1

1

1

F r o m t h i s l a t t e r o b s e r v a t i o n the q u e s t i o n i m m e d i a t e l y arises as t o t h e p o s s i b i l i t y of a p p l y i n g E q u a t i o n 22 to channels where there is a sizable gap between t h e c h a n n e l a n d t h e floor. I n a d d i t i o n one m i g h t ask t o w h a t d e gree t h e e q u a t i o n w o u l d be a p p l i c a b l e t o t h e a p p a r a t u s of D a v i e s (1), where t h e c h a n n e l d e p t h is e x t r e m e l y s m a l l since i t is f o r m e d b y t h e contact of rings w i t h t h e l i q u i d surface. T a b l e I I i l l u s t r a t e s t h e a p p l i c a t i o n of E q u a t i o n 22 to e x p e r i m e n t a l d a t a , where t h e r e p o r t e d surface v i s c o s i t y was measured b y t h e m e t h o d of D e r v i c i a n a n d J o l y (3), a n d t h e p a r t i c l e t i m e s were measured b y D a v i e s . I n t e r m s of p a r t i c l e t i m e E q u a t i o n 22 becomes

M

S

=

—

— j y -

(29)

T h e substrate was w a t e r i n a l l cases, t h u s p e r m i t t i n g μ to be e s t i m a t e d as 1 cp. T h e reported c h a n n e l w i d t h was 3.5 m m . , a n d t h e p a r t i c l e t i m e s for clean w a t e r p r o v i d e d t *. E q u a t i o n 29 has been a p p l i e d t o give t h e c a l c u l a t e d surface v i s c o s i t y . ( F o r each f i l m , surface c o n c e n t r a t i o n is i n d i 6

c


22.

BURTON

AND

Table II.

Reported Surface Viscosity, Surface Poises

Clean water Stearic acid, A . on 0.01ΛΓ HC1 20.1 A . on 0.01ΛΓ HC1 20.1 A . on O.OOliV HC1 Octadecanol, 20.8 A . on O.OliV HC1 2

2

2

2

Rheological

327

Measurements

Application of Equation 29 to Calculation of Surface Viscosity from Davies' Measurements (2)

Material


Surface

M A N N H E i M E R

0

c

Calculated Surface Viscosity, Surface Poises 0

78

2.3 Χ 10" 6.3 Χ 10" 12 X 10~

4

4

4

44 Χ

Reported Particle Time, t Seconds

10"

4

105 130 161 300

3.8 Χ 10" 7.4 X 10~ 12 Χ ΙΟ"

4

4

4

32 Χ

10"

4

cated as A . per molecule.) T h e r e was considerable scatter i n t h e m e a s u r e d p a r t i c l e t i m e , some u n c e r t a i n t y i n t h e r e p o r t e d surface v i s c o s i t y , a n d c o n siderable q u e s t i o n as to the a p p l i c a b i l i t y of E q u a t i o n 29 to t h e zero d e p t h c h a n n e l . I n v i e w of a l l t h i s , i t is g r a t i f y i n g t o see t h e agreement between c a l c u l a t e d a n d r e p o r t e d surface viscosities. S i n c e t h i s a p p a r a t u s g e o m e t r y represents a n extreme case, i t s h o u l d give rise to confidence i n a p p l i c a t i o n of E q u a t i o n 29 to channels m o r e n e a r l y l i k e t h a t u p o n w h i c h the d e r i v a t i o n was based. 2

Conclusions A n arrangement s u c h as s h o w n i n F i g u r e s 2 a n d 3 c a n serve as a s u r face viscometer, a n d i t is a m e n a b l e to r e l a t i v e l y precise a n a l y t i c a l t r e a t m e n t . I t s chief a d v a n t a g e is i n t h e s i m p l i c i t y of t h e u l t i m a t e equations for surface v i s c o s i t y ( E q u a t i o n s 2 1 , 22, a n d 29), a n d for surface stress i n r i g i d film ( E q u a t i o n 28). T h o u g h i t is d e m o n s t r a t e d t h a t E q u a t i o n 29 applies also as a n a p p r o x i m a t i o n i n D a v i e s ' earlier c o n f i g u r a t i o n , t h i s s h o u l d n o t suggest a b a n d o n m e n t of t h e advantages of t h e r e c o m m e n d e d deepc h a n n e l geometry. I n the o p i n i o n of the w r i t e r s t h e deep c h a n n e l offers as basiC., or absolute, a n a p p r o a c h to t h e d i r e c t measurement of surface v i s c o s i t y as does t h e c o n f i g u r a t i o n used b y J o l y a n d D e r v i c i a n , a n d at t h e same t i m e i t is a p p l i c a b l e t o soluble films. I n d e e d , t h e " s e l f c o r r e c t i n g " n a t u r e of E q u a t i o n 20 appears t o m a k e i t t h e more accurate a p p r o a c h to o b t a i n i n g absolute values of μ . W o r k continues o n d e v e l o p i n g equations for i n t e r f a c i a l v i s c o s i t y , where b o t h fluids are l i q u i d s , a n d for n o n - N e w t o n i a n surface b e h a v i o r , where μ v a r i e s w i t h stress. I n a l l s u c h cases t h e deep c h a n n e l g e o m e t r y consider a b l y simplifies these d e r i v a t i o n s . A s these analyses are developed, t h e p r o m i s e of t h i s arrangement as a basic i n s t r u m e n t s h o u l d i m p r o v e . 8

8

Nomenclature D = w=

channel d e p t h - w i d t h ratio velocity magnitude i n ζ direction


328

ORDERED FLUIDS AND LIQUID CRYSTALS

w = mid-channel velocity magnitude in free surface of liquid w* = mid-channel velocity magnitude in free surface of liquid without sur face viscosity Wf— velocity magnitude of channel floor χ = coordinate measured upward from channel floor xo = channel depth X= dimensionless coordinate, χπ/yo y = coordinate measured across channel y = channel width F = dimensionless coordinate, yir/yo z= coordinate measured along channel μ = bulk viscosity of liquid μ = surface viscosity of liquid v= modified bulk viscosity, ΐ/ομ&Ατ τ = shear stress on surface T s = yield value of shear stress for rigid films s= subscript, refers to liquid surface δ = thickness of surface region of enhanced viscosity c

0

6


8

8

Literature

Cited

(1) Davies, J. T., Proc. Second Intern. Congr. Surface Activity 1, 220 (1957). (2) Davies, J. T., Rideal, Ε. K., "Interfacial Phenomena,'' 2nd ed., p. 258, Academic Press, New York and London, 1963. (3) Dervician, D. G., Joly, M . , Compt. Rend. 204, 1318 (1937). (4) Ewers, W. E., Sack, R. Α., Australian J. Chem. 17, 40 (1954). (5) Harkins, W. D., Kirkwood, J. Q., J. Chem. Phys. 6, 53 (1938). (6) Joly, M . , "Recent Progress in Surface Science," p. 1, Academic Press, New York and London, 1964. (7) Plateau, J. A. F., Phil. Mag., Ser. 4, 38, 445 (1869). RECEIVED February 11,1966. Preliminaryfindingsof the U. S. Army Fuels and Lubri cants Laboratory, Southwest Research Institute, San Antonio, Tex., currently conduct ing a program to determine the effect of surface rheology on oil foaming.


Ordered Fluids and Liquid Crystals

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