유변학

유변학

점탄성

점도

- 점도 계수 (viscosity coefficient)

- 제 1수직응력차 계수 (first normal stress difference coefficient)

- 제 2수직응력차 계수 (second normal stress difference coefficient)

물질함수

물질함수







점도

❏ The units are . . .





 10 Poise =1 Pa

.

1 cP (centipoise) = 1 m Pa^.s

단순전단유동

Shear Stress:  = F/A Strain:  = dx/y

Shear Rate:  = d/dt = V/y

Viscosity: 

y x

x

dx

F = Force V = speed

y

A = area

그림 2. 뉴튼성유체의 전단응력과 전단율과의 관계(Denn, 1980).

점도

1000

1.000E-6 1.000E-4 0.01000 1.000 10.00 100.0

shear rate (1/s)

10000

1.000E-3 0.01000 0.1000 1.000 10.00 100.0 1000

viscosity (Pa.s)

Xanthan/Gellan Fructose Soln.

N450,000

S3

점도

TA Instruments

그림 3. LDPE의 서로 다른 온도에서의 점도(Bird et al., 1987).

전단박화

Unsheared Sheared

Aggregates break up

Random coil

Polymers elongate

Anisotropic Particles align with the Flow Streamlines

~ 1 s

전단박화

TA Instruments

(7)

(8)

(9) Cross equation:

Carreau model:

Power-law model:

점도모델

전단후화

1000

0.0100 0.100 1.00 10.0 100.0

shear rate (1/s)

0.3500

0 0.05000

0.1000 0.1500 0.2000 0.2500 0.3000

viscosity (Pa.s)

Flip Chip Underfill Resin + Filler [var. %]

@ 80°C

0%

35%

65%

70%

전단후화

TA Instruments

항복응력

마스터곡선

전단속도(1/s)



Sedimentation 10



Molecular Structure



Leveling/Sagging 10

to 10



Compression Molding



Pouring 10

to 10



Extrusion



Pumping 10

to 10



Blow Molding



Rubbing 10

to 10



Injection Molding



Spraying 10



Bearing lubrication 10

점도 (Pa.s)

Asphalt Binder ---

Polymer Melt ---

Molasses ---

Liquid Honey ---

Glycerol ---

Olive Oil ---

Water ---

Air ---

100,000 1,000 100 10 1

0.01

0.001

0.00001

점도

• Relative viscosity

• Specific viscosity

• Reduced viscosity

• Inherent viscosity

• Intrinsic viscosity

s

rel



  

s s

sp









c

sp red

  

c

rel inh

  ^ln 

  c c

rel c

sp c

 

 lim lim ^ln

0

0 







rel

/ c ln 

sp

/ c



  ^{  KM}

고유점도

브룩필드 점도기

선형점탄성(linear viscoelasticity)

변형

0 시간

응력

0 시간

응력

0 시간 응력

0 시간

물질의 완화특성

( , ) ( , )

t

G t   





^ _

/

( ) 0 ^t

G t

G e

^{G t ( ) }  ^{G e}

1. coordinate invariant 2. material objectivity (frame indifference)

GNF model

I=0 for incompressible fluid III=0 for simple shear flow

power-law model

Carreau-Yasuda model

Bingham model

Pressure-driven flow

constitutive equations

Maxwell model

Generalized Maxwell Model

/ 1

( ) ^k

N k t

k k

G t  e ^











GLVE model

1 2

0 G s ds( ) , 0

 



선형점탄성 모델

e v

    

     

 

E E E E

   E

 

   

SAOS

SAOS

d dt

 





0G'sin t 0G"cos t

        0 cos t  0 cost

2

0 0

2 2

( )

' , "

1 ( ) 1 ( )

G G

G  G 

 

 

 

2

2 2

( )

' , "

1 ( ) 1 ( )

k k

k k

k k

G G   G G  

   

 

 

 

/ 0

( ) H( ) ^s

G s  e ^d



 





2 0 2

0

( )

( ) 1 ( )

4

L z

P P R r

v r L R

  

   

limitations of GLVE model

material objectivity (frame indifference)

0 G s( ) cos(2 s ds)

 



valid only for slow flow

물질함수

steady shear

( )

t





21 0

1 11 22

1 2 2

0 0

22 33

2

2 2 2

0 0

( )

( )

( )

( )

( )

N

N

  



 

  

 

  

 

  

  

 

   

stress growth

0

0 0

( ) 0

t t

 t



 

  

21 0

11 22

1 2

0

22 33

2 2

0

( , )

( )

( , )

( )

( , ) t

t

t

  



 

 

 

 







 

 

 

 

 

stress relaxation

0 0

( ) 0 0

t t

t





 

21 0

11 22

1 2

0

22 33

2 2

0

( , )

( )

( , )

( )

( , ) t

t

t

  



 

 

 

 







 

 

 

 

 

creep

21

0

0 0

( ) 0

t t

 t



 

  

21 0

0

(0, )

( , )

t

J t  







0

0

( ) _s

t

J t J

 



step strain

0 0

0 0

( ) lim 0

0

t

t t

t



 





 

   

 

 ¹

2

21 0

0

0

11 22

0 2

0

22 33

0 2

0

( , ) ( , )

( )

( , )

( )

( , ) G t t

G t

G t

 

 

 

 

 

 





 

 



 

