1. 고분자 물성
(자료 8)
울산대학교 화학과
정 한 모

2. Chapter ⅩⅧ. Linear Viscoelasticity
Die Swell
The Weissenberg effect

3. 2. Mechanical models for linear viscoelastic response
Constitutive equation
a) Elastic (Spring) : The elastic deformation is instantaneous
and independent of time
= G ∙
Shear stress
shear strain
shear modulus Or
= E ∙
Tensile stress
tensile strain
tensile modulus
G ∙ 2 ( ) = E
Poisson’s ratio:
for rubber = 0.5
E = 3G

4. b) Viscose (Dashpot) = ∙ Or = ∙ Normally = 3 shear viscosity Newtonian
fluid Or
= ∙
shear viscosity
= ∙
tensile viscosity
Normally
= 3
however there are
many exceptions

5. C) Maxwell model = G = ∙ = = = + ` = ∙ + ∙ • Or = - ∙ = - (18.5)
= ∙
= = =
양변 미분하면 `
= ∙ ∙ Or
= ∙ •
= (18.5)
relaxation time
N ∙ s/m2
N/m2
= sec

6. d ) Voigt – Kelvin model
= = =
∴ = (18.10)

7. 2) Creep : A constant stress is instantaneously applied
to the material, and the resulting strain is followed as
a function of time
Spring
= G
if constant is applied =

8. b) Dashpot
= ∙
dt = ∙
= (constant)
t = ∙ C
at t = 0,
= 0 ⇒ C = 0 ∴
t = ∙
= ∙ t

9. C) Maxwell model
elastic strain
creep
recovery
permanent set

10. = ∙ ∙
= (constant),
= 0 =
= ∙ t + C
at t = 0,
= ⇒ C =
∴ = ∙ t (18. 6a)
Because, compliance J = =
Creep compliance JC (t) =
JC (t) = (18. 6b)

11. d) Voigh – Kelvin model
= (18.10)
= (constant) =
In order to solve this differential equation,
both sides were multiplied by e
Integrating both sides;

12. at t = 0 = 0 (because of dashpot)
= (1 – e ) (18. 11a) Or
JC (t) = = (1 - e ) (18. 11b)

13. If external force is released at t = ts
= 0,
= C x 1 ⇒ C =
= ∙ e
From previous equation (A)
= C ∙ e

14. Example) When 250 N was applied on the following PP cube;
200 mm
25 mm
3 mm
After 100 sec the length extended to mm Calculate
tensile creep compliance at this time.
Solution :
D =
tensile creep compliance
250N
(25 x 10-3m x 3 x 10-3m)
= 3.33 x 106 Pa
(t = 100sec) ∆l lo =
0.5 x 10-3m
200 x 10-3m
2.50 x 10-3
∴ D =
3.33 x 106 Pa
= 7.51 x Pa-1
= GPa-1 =

15. 3) Stress relaxation : Suddenly applying a strain to
the sample and following the stress as a function
of time as the strain is held constant.
Spring
= G ∙
if = (constant)
time G

16. b) Maxwell model

17. = ∙ ∙
= (constant) and = 0
∙ = 0
= C ∙ e
at t = 0,
= G
∴ = G ∙ e
= G ∙ e

18. Example 1. Examine the response of a Maxwell element
in an engineering stress-strain test, a test in which the
rate of tensile strain is maintained (approximately)
constant at ∙
Solution)
= (18.5)
Similarly, in the case of tensile stress, it can be expressed
= ∙
If this differential equation is solved with initial condition;
= at t = 0
= [1 – e ]
Now = (constant)
and = at t = 0
Therefore = t ⇒ t =
Then
= [1 – e ]

19. • Modulus is independent of (or )
for Hookean solid. However, modulus increases
as strain rate is increased for viscoelastic solid. ∙

20. 3) There parameter model
( ) = ( ) (18.13) ∙ or = = ∙

21. 4) Four parameter model: Creep
• More parameters ⇒ More accurate description of the
viscoelastic behavior of polymer
• Creep behavior
= , = 0
= at t = 0
= [1 – e ] (18. 16a)
spring
dashpot
Voigh- Kelvin
• Creep response is the sum of the creep response of
the Maxwell and Voigt-Kelvin elements

22. Instantaneous elastic strain : Spring 1
elastic strain of bond angles and lengths
Retarded elastic strain (Dashpot 2, V-K)
Retarded elastic recovery (Spring 2, V-K)
Cooperative uncoiling and coiling
Equilibrium viscous flow (Dashpot 1)
Instantaneous elastic recovery (Spring 1)
Permanent set (Dashpot 1)

23. Example 2. Using the four-parameter model as a basis, sketch
qualitatively the effects of (a) increasing molecular weight and
(b) increasing degrees of crosslinking on the creep response
of a linear, amorphous polymer.
Solution) a. As discussed in Chapter XV, The equilibrium zero-
shear (linear) viscosity of polymers, represented by η1 in the
model, increases with the 3.4 power of Mw. Thus, the slope in
the equilibrium flow region τ0/η1 is greatly decreased as the
molecular weight increases, and the permanent set (τ0/η1)ts is
reduced correspondingly (Fig. 18.9).

24. b. Light crosslinking represents the limit of case (a) above,
when the molecular weight reaches infinity, since all the
chains are hooked together by crosslinks. Under these
conditions, they can’t slip past one another, so η1 becomes
infinite. If the crosslinking is light (crosslinks few and far
between), as in a rubber band, coiling and uncoiling won’t be
hindered appreciably. Note that crosslinking converts the
material from a fluid to a solid (it eventually reaches an
equilibrium strain under the application of a constant stress)
and it eliminates permanent set. The equilibrium modulus will
be on the order of 106 to 107 dyn/cm2 (105 to 106 N/m2=Pa),
the characteristic “rubbery” modulus. Further crosslinking
begins to hinder the ability of the chains to uncoil and raises
the restoring force (increases η1 and G2). At high degrees of
crosslinking, as in hard rubber (ebonite), the only response
mechanism left is straining bond angles and lengths, giving
rise to an almost perfectly elastic material with a modulus on
the order of 1010 to 1011 dyn/cm2 (109 to 1010 N/m2), the
characteristic “glassy” modulus.

25. Stress relaxation for Maxwell model : λc = (where : λ = ) (18.7a)
5) Deborah number : De
Characteristic time (λc) : The time required the material to reach (63.2%) of its ultimate retarded elastic response to a step change
Stress relaxation for Maxwell model : λc =
(where : λ = ) (18.7a)
at t=λ, 1 e η G η G

26. Creep for Voight-Kelvin model : λc= (18.11a) at t=λ,η G

27. If the stress is suddenly removed from a rotational viscometer, the creep recovery or elastic recoil of the material can be followed. This provides a value of λc for the material.
viscosity ↓ ⇒ λc ↓

28. λc is shorter than pass time ⇒ viscous
De =
De>>1 elastic
De → 0 viscose
Example 4. A paper cup containing water is placed on a stump. A .22-cal bullet fired at the cup passes cleanly through, leaving the cup sitting on the stump. The water is replaced by a dilute polymer solution in a second cup. This time the bullet knocks the cup 25 ft beyond the stump. Explain.
For a polymer such as glass
If ts is short De >>1 ⇒ elastic
If ts is long De <<1 ⇒ viscous λc ts
Time of deformation
bullet
pass
λc is shorter than pass time ⇒ viscous
bullet
bounce
λc is longer than pass time ⇒ elastic

29. time dependent relaxation modulus
6) Generalized Maxwell model
Stress relaxation for each Maxwell model
(18.7a) (18.18)
Stress relaxation for generalized Maxwell model
(18.19a)
(18.19b)
If n is large
(18.20)
time dependent relaxation modulus
continuous distribution
of modulus vs relaxation time

30. The G(t) function curve does not have to run exactly through each of the discrete spectrum as presented in the figure.
Rheologist find G(λ) of a material and utilize it.
For example MWD can be determined from G(λ). * *

31. 3. The Boltzmann superposition principle
In creep-behavior;
Strain response due to complex stress loading is the
linear additive of the strains due to each step
Total final deformation is the sum of each
contribution
In stress relaxation behavior;
Stress responses due to complex strain loading
• The stresses resulting from each individual strain are
linearly additive

32. time-dependent relaxation modulus2
∆τ0
∆τ1
∆τ2
(18-30)

33. Example 5. Maxwell element is initially free of stress and strain
Example 5. Maxwell element is initially free of stress and strain. At time t=0, a train of magnitude is suddenly applied and maintained constant until t=λ/2, at which time the strain is suddenly reversed to a value of - and maintained at that value (Fig a). Obtain an expression for τ(t) and plot the result.
Solution) For Maxwell model
(18.7a)
This term is valid
only when t>t1= λ 2

34. Generally
(18.31)
Example 1. Examine the response of a Maxwell element in an engineering stress-strain test, a test in which the rate of tensile strain is maintained (approximately) constant at ε0.
Example 6. Solve Example 1 by applying the Boltzmann Superposition Principle, thereby demonstrating how stress-time response in an engineering stress-strain test may be predicted from stress-relaxation data. . .

35. . . . . Solution) For Maxwell model the stress relaxation by tensile
deformation ε0 ;
From (18.31) ε0 . . t . .
The same result as
in Example 1 (P 304)

36. Example 7. To demonstrate the fact that the stress in a viscoelastic material depends on its past strain history, calculate the stress τ(ts) in a Maxwell element initially free of stress and strain that is brought to a strain at time ts by three different path :
a. For t’ < 0, = 0 ; for 0 ≤ t’ ≤ ts, = .
b. For t’ < ts, = 0 ; for t’ ≥ ts, = . c
Solution)
a. This is good old stress relaxation. From (18.7a)
b. This corresponds to the initial extension in stress relaxation :
c. This is the shear analog of an engineering stress-strain test with the constant shear rate By analogy to the solution of Example 6 above . . .

37. 4. Time-temperature superposition
Applicable to any viscoelastic response
(creep. stress relaxation, etc)
short time at high T = long time at low T
For stress relaxation

38. Curves can be shifted along the time(x) axis to partially
overlap, giving a master curve
aT= : shift factor
Curve at 135 ℃ can partially overlap with curve at 115 ℃ (reference temperature) to make smooth cure when shifted 3 order :
∴ aT=10-3=
This means that 10-3 time is necessary at 135 ℃ compared to that at 115 ℃ tT
tT0
t135
t115
At 100 ℃ : the shift along x axis to make master
curve is 5 order
∴ aT=105=
At 40 ℃ : aT=1010=
t100
t115
t40
t115

39. Five regions of viscoelastic behavior
a) Glassy region
∙ At low temperature or short time (large De)
∙ only bond angles and lengths can respond
∙ glassy modulus (109~1010 Pa)
b) Leathery region
c) Rubbery plateau
∙ Uncoiling of the chains
∙ rubbery modulus (105~106 Pa)
d) Rubbery flow region
∙ Still quite elastic but has a significant flow
component
e)Viscous flow region
∙ Molecular slippage

40. Molecular weight have no significant influence on straining of bond angles and lengths or on uncoiling.
So, the glassy and rubbery moduli are uncharged.
Flow is severely retarded by increasing molecular weight, which extends the rubbery plateau. For very low molecular weight polymers, the rubbery plateau isn’t even seen.
In the limit of infinite molecular weight (light crosslinking), flow is entirely eliminated, and the curve levels off with the rubbery modulus.
Higher degree of crosslinking restricts uncoiling, ultimately leading to a material that responds only by straining of bond angles and lengths.

42. -C1(T-T*) C2+(T-T*) WLF equation log at = (18.79) T* C1 C2Tg
Tg+(50 ± 5)
Example 9. The master curve for the polyisobutylene
in Fig indicates that stress relaxes to a modulus
of 106 dyn/cm2 in about 10h at 25 ℃. Using the WLF
equation, estimate the time it will take to reach the
same modulus at a temperature of -20 ℃. For PIB,
Tg = -70 ℃
Solution. To use the WLF equation, the reference
temperature must be Tg = -70 ℃:
-C1(T-T*)
C2+(T-T*)

43. 5. Dynamic mechanical testing (Oscillatory test)
Basic principles
● Two-plate-model
• ± = ± F/A
± = ± s/h = ± tan
• = 2 f
Angular frequency (1/s)
ex) f = 10 Hz, = 62.8 s-1
frequency (Hz : 1/s)

44. ● For ideal elastic materials
• (t) curves is always in phase with (t) curve.
(t) = max sin t
(t) (t) = G max sin t

45. ● For ideal viscous fluids
• A delay of (t) relative to (t) ;
phase shift angle 90 ( )
(t) = sin t
(t) = ∙
= ∙ ∙ cos t
= ∙ ∙ ( t+ )

46. ● For viscoelastic materials
• A delay of (t) relative to (t) ;
phase shift angle (0≤ ≤ )
(t) = sin t
(t) = sin ( t + )

48. ● This can be thought of as being a projection of
two vectors, and , rotating in the complex
plane.

49. → → • When the applied strain is the independent variable
• Storage modulus :
• Loss modulus :
tan = = → →

50. • Complex modulus :

51. It reveals the ratio of the viscous to the elastic portion
of the deformation behavior.

52. ● Reaching tan = 1 is an important analysis criteria
in hardening or gel formation processes
(sol/gel transition point).
In the liquid state, tan >1
(because G”>G’), and in the gel state,
tan <1 (because G’>G”)

53. ● The tack behavior can be evaluated and controlled
using the loss factor (tan ). The sample shows
stringiness only if the tan value is in a medium range.
Thus, tackiness can be reduced or even prevented if
tan values are produced to be either lower or higher
than the values in this medium range.

54. • = 90° or tan = ∞
Ideal viscous flow behavior
• G” > G’
Behavior of a viscoelastic liquid
• =45° or tan = 1 or G’ = G”
viscous-elastic behavior with a 50/50 ratio of
viscous/elastic portions
• G’ > G”
behavior of a viscoelastic solid
• = 0° or tan = 0
Ideal elastic behavior

55. → ● Work done on a unit volume of material undergoing
a pure shear deformation.
(14.3) (work per unit volume)
(18.46)
(18.47)
(18.49) →
Here,
and

56. • Energy dissipated per cycle :
for
(18.51a)
energy stored
elastically
viscous dissipation; energy converted to heat through molecular friction
Stored elastic energy is recovered
• Energy dissipated per cycle :
(18.52)
• Energy dissipated per time :
(18.53)

57. 2) Measurement of dynamic properties by torsion
pendulum (free oscillation)
• Damping : •
∆ = ln = ln = ln (18.60)
∆ = (18.61) ~
• G’K” = I (1+ ( )2 ]
geometric factor
for specimen
for thin wall tube
moment of inertia of oscillating bar
K”=
length
wall thickness
Average
diameter
( : period of oscillation)

58. Example) The restoring element is a piece of plasticized
vinyl laboratory tubing and the formula for a thin-walled
tube is used.
I = 7.4 x 103 g∙cm2 , L = 10 cm, t = 0.15 cm
D = 0.85 cm, oscillation period = 1.67 sec
∆ = 0.23
Solution;
K” =

59. 3) Measuring modes
(1) Time sweep
a. Gel point
• at fixed and
for example ;
• usually represented in a semi-logarithmic
diagram

60. b. Thixotropic behavior
Reversible structure decomposition and regeneration
G” value changes little compared to G’ value

61. (2) Temperature sweep: 보통 고체 상태에서 측정
• Usually represented in a semi-logarithmic
• 보통 1 Hz 부근에서 측정
a. Amorphous polymer
• Normally, α-relaxation at the highest temperature.
β-relaxation at the next highest temperature
and so on
• Normally, α-peak is Tg for amorphous polymer.
(Sometimes, G” peak or G’ decrease temperature)
• β-peak, peak, and so on ; side
chain motion or partial motion of main chain

62. • Glassy modulus : 1 GPa
Rubbery modulus : 1 MPa
• 보통 tan ≅ 0.01
in such a case
ㅣG*ㅣ ≅ G′ ≅ G
• tan peak temperature depends on imposed frequency
Ea : Activation energy for motion.
In the case of side
chain motion of PMMA; Ea = 17 kcal/mol

63. ● In terms of four parameter model;
• Below Tg :
∙ Reversible motion of few atoms
∙ No translational motion
∙ Almost completely elastic
∙ Only spring 1 is operative ( , )
• In the vicinity of Tg
∙ Translational motion begins
cannot
pass
can hardly
can easily
below Tg
Tg vicinity
Reversible
translation
above Tg
∙ drops to the point where it can deform and
dissipate energy, giving damping
∙ High G1 and do not respond

64. • Above Tg
∙ Reversible and easy translational motion (marginal
damping) ⇒ elastic
∙ Physical or chemical crosslink persist
∙ ↓
∙ Mainly spring 2 respond (G2 < G1)
• At still higher temperature
∙ Physical crosslink can not persist
∙ Viscous flow
∙ G2 also decrease

65. b. Crystalline polymer
4.25 Poly(ethylene terephthalate) when quenched from the melt has 0% crystallinity; when it
is heated through Tg the modulus drops dramatically as indicated. Specimens of crystallinity
26%, 33%, and 40% have modulus-temperature plots as shown. These three values of
crystallinity were produced by produced by heating quenched specimens at the temperatures
Tc prior to the temperature scan (after llers and Breuer).

66. ● Greater number of mechanical relaxation in crystalline polymers: PE
4.38 Limited mobility in the amorphous region of polyethylene. The -mechanism (see Figure 4.12) is attributed to movements of this type (‘crankshaft rotation’).

67. ● Effect of water: polyurethane
• two major relaxation above room temperature
(glass-rubber transition and melting)
and two smaller ones below
• Water content ↑
∙ Tg↓
∙ anti-plasticizing effect below Tg
(in the case of nylon and polyurethane)
4.26 Temperature scan of G’ and Λ for a dry polyurethane with structure
[ (NH)-(CH2)u-(NH)-(CO)-O(CH2)v-O-(CO) ]n
with u=6 and v=4 (after Wolf and Schmieder).

68. ● Effect of crosslinking

69. (3) Frequency sweep: 보통 액체 상태에서 측정
a. Linear viscoelastic (LVE) range
∙ Normally at
∙ ln most cases the G’( ) function is taken for
analysis because the G’ curve almost always
fall before the G” curve.

70. b. Frequency sweep of Maxwell model
a) Single Maxwell model : narrow MWD
lf we solve the differential equation (p 326)

71. Curve discussion for polymers with a narrow MWD (see Fig. 8.12):
1) In the log/log diagram, the G’( ) curve rise with the slope 2:1
at low frequencies, because G’ ~ 2 for → 0.
2) At high frequencies (with → ∞), the G’( ) curve reaches the
constant plateau value GP
A Maxwell liquid is inflexible and rigid under very rapid movements,
and the parameter GP represents the maximum rigidity under this
condition.
Result for the frequency-dependent limiting values of the storage
modulus: limω→0 G’( ) = 0, and limω→∞G’( ) = GP
3) In the log/log diagram, the G”( ) curve rises with the slope 1:1 at
low frequencies, because G”~ for → 0.
4) At high frequencies, the G”( ) curve falls with the slope (-1): 1,
because: G” ~ 1/ for →∞.
5) At the point ∙ =1, the G”( ) curve reaches its maximum with
the value G”max = GP/2. If the period of the oscillation is 2
(or if 1/ = , respectively), then the viscous and elastic behavior
are the same size (G’ = G” = GP / 2).
Result for the frequency-dependent limiting value of the loss modulus:
limω→0 G”( ) = 0, and limω→∞G”( ) = 0

72. b) Generalized Maxwell model : wide MWD
( single relaxation time vs a spectrum of
several relaxation time)
Initial range (or flow zone 1) at ≤ 1: At low frequencies, the behavior
can be described using a single Maxwell model. Here, the G’( ) and G”( )
curve have a slope of 2:1 and 1:1, respectively(as in Fig. 8.12).
The polymer therefore displays viscoelastic behavior with a dominant
viscous portion (G”>G’), although the sample can be highly viscous.
2) Rubber-elastic range with the plateau value GRP between 1 and 2
: In the medium frequency range, the so-called “rubber-elastic plateau”
occurs with the value GRP. Hear, the longer molecular can no longer
glide along each other and all remaining entanglements are now part
of a temporary network. The polymer displays viscoelastic behavior
here with a weakly dominant elastic portion, i.e. G’>G”.
Fig. 8.13: G’( ) and G”( ) of two polymers with different MWD, showing the plateau
value GP and the rubber-elastic plateau value GRP
A polymer with two fractions, each with a very narrow MMD, showing the plateau
GP and the values. (In reality such samples are very rare.)
A polymer with a wide MMD

73. 3) Transition range (or flow zone 2) between 2 and 3:
G’( ) and G”( ) increase again more steeply. In this range only the
smaller and more mobile molecules are still deformable, e.g. oligomers
and softening agents or limited parts of the macromolecules.
The elastic portion becomes increasingly dominant.
4) Glassy range with the plateau value GP at ≥ 3:
G’( ) reaches a constantly high value and G”( ) decrease. Here, movement
between the molecules chains is no longer possible because the more flexible
parts of the macromolecules and the smaller molecules are now immoblie in
this state. Their behavior can be imagined as in a “frozen condition”.
The molecule chains may vibrate a little but can no longer follow the rapid
oscillation movements. The polymer now displays the behavior of a rigid solid
with a clearly dominant elastic portion

74. c) Effect of MW and MWD
1) A comparatively high plateau value GP indicates a higher degree of structural
strength of the temporary network when exposed to rapid movements.
This can be interpreted as follows: There is a greater amount of entanglements
which indicates that longer molecules with a higher molar mass are present.
2) If the curves are shifted towards lower frequencies, this indicates longer or
highly branched molecules can still move against each other at low frequencies
but begin to block each other more and more as the frequency increases.
This causes the G’ and G”: curves to rise.
When even higher frequencies are reached, the smaller molecules also begin
to block each other. Therefore, if the curves are shifted towards higher
frequencies, this indicates the presence of comparatively small and/or less
branched macromolecules.
3) A relatively steep rise in the curves is the result of a narrow MWD.
A wider MWD results in a less steep G’( ) curve with a more gradual transition
to the plateau value GP (see Fig.8.16), and in a G”( ) curve which has a less
pronounced peak.
4) When comparing two polymers with the same average molar mass but different
MWD, the following applies: the crossover point G’ = G” occurs at a lower G
value for the polymer with the wider MWD.
5) When comparing two polymers with the same MWD but different average molar
masses, the following applies: the crossover point G’ = G” is found at a lower
angular frequency CO for the polymer with the higher average molar mass
because its molecules are less flexible and less mobile (see Fig. 8.14, the index
“co” in CO comes from “cross-over point”)
Fig. 8.14:
Horizontal shift of the crossover point of
G’( ) and G”( ), dependent on the
average molar mass M (here: M1 > M2)

75. d) Crosslinked polymers
Fig. 8.15:
G’( ) and G”( ) of two cross-linked
polymer
Sparsely cross-linked polymer
(flexible at rest, e.g. elastomer),
Densely cross-linked polymer
(rigid, e.g. thermosetting plastic)
Fig. 8.16:
Comparison of G’( ) curves for samples
With different structures
Polymer with unlinked molecules and
a narrow MMD,
(2) Polymer with unlinked molecules
and a wide MMD,
(3) Sparsely cross-linked polymer, flexi-
ble gel or dispersion with low structural
strength at rest,
(4) Densely cross-linked polymer, rigid
gel or dispersion with high structural
Strength at rest.

76. ● Cox/Merz rule •
• found empirically in 1958
at (18.56)
• This rule is especially useful when only mechanical
interactions in the molecular structure are responsible
for the rheological behavior, as for example entangled
macromolecules, Any kind of physical and/or chemical
interactions lead to deviation from the Cox/Merz rule.