고분자 물성 (자료 8) 울산대학교 화학과 정 한 모

Chapter ⅩⅧ. Linear Viscoelasticity Die Swell The Weissenberg effect

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 (1 + ) = E Poisson’s ratio: for rubber = 0.5 E = 3G

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

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

d ) Voigt – Kelvin model = = = + ∴ = + (18.10)

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 =

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

C) Maxwell model elastic strain creep recovery permanent set

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

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

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

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

Example) When 250 N was applied on the following PP cube; 200 mm 25 mm 3 mm After 100 sec the length extended to 200.5 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 10-10 Pa-1 = 0.751 GPa-1 =

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

b) Maxwell model

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

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; = 0 at t = 0 = [1 – e ] Now = (constant) and = 0 at t = 0 Therefore = t ⇒ t = Then = [1 – e ]

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

3) There parameter model (1 + ) = ( 1+ ) (18.13) ∙ or + = + = ∙

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

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)

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).

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.

Stress relaxation for Maxwell model : λc = (where : λ = ) (18.7a) 5) Deborah number : De Characteristic time (λc) : The time required the material to reach 1- (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

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

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 ↓

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

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

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). * *

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

time-dependent relaxation modulus 2 ∆τ0 ∆τ1 ∆τ2 (18-30)

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. 18.13a). 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

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. . .

. . . . 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)

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 . . .

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

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

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

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.

-C1(T-T*) C2+(T-T*) WLF equation log at = (18.79) T* C1 C2 Tg 17.44 51.6 Tg+(50 ± 5) 8.86 101.6 Example 9. The master curve for the polyisobutylene in Fig. 18.21 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*)

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)

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

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

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

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

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

• Complex modulus :

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

● 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”)

● 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.

• = 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

→ ● 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

• 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)

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)

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” =

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

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

(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

• 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

● 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

• 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

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).

● 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’).

● 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).

● Effect of crosslinking

(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.

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

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

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

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

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)

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.

● 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.