Chapter IV. Analysis of Research Topics and Scientific Collaborations in Energy
IV- 7. References
It is now well known that the deformation of polymers during the flow has dramatic effects on the crystallization and subsequent solidification of polymers in two aspects: enhancement of crystallization rate and formation of thread-like nuclei that grow into shish-kebab structures if the deformation of the fluid is sufficiently strong (Zhu and Edward [428]). The two phe-nomena are known as the flow-induced crystallization. Early investigations in this field can be traced back to around the early 1970s (Hass and Maxwell [147]; Mackley and Keller [236]). In a vast literature, for useful results towards the prediction of structure formation during injection molding, one should especially note the work of the polymer research group at University of Linz (Eder and Janeschitz-Kriegl [91]; Janeschitz-Kriegl [181]).
In this book we mainly consider the post-shear crystallization of polymers, that is, crystalliza-tion occurring after the material experienced a short-term shearing. The short-term shearing
150 10 Crystallization
treatment is relevant to the injection-molding process. The reader who is interested in crystal-lization under continuous shearing is referred to the work of Hadinata et al. [140].
10.2.1 Enhanced Nucleation
Eder et al. [92] have suggested the following first-order differential equation for the flow-enhanced nucleation:
N˙f+ 1 λN
Nf = f , (10.24)
where Nf is the flow-induced nuclei number density,λN is a temperature-dependent relax-ation time that usually takes a large number, and f is a function depending on flow variables and temperature. Eder et al. consider the shear rate as a driving force for the flow-enhanced nucleation, and write f in the following form:
f = gn
µγ˙ γ˙n
¶2
, (10.25)
where gn is a factor with the dimension s−1m−3, and ˙γn is a shear rate of activation. The quadratic dependence on ˙γ2 was introduced originally based on the argument that the nu-cleation process should not depend on the shearing direction. Janeschitz-Kriegl [181] later interpreted this term as the specific work.
Several other assumptions on the driving force have been adopted by different authors.
Zuidema et al. [435] replaced the ˙γ2 term by the second invariant of the deviatoric part of the recoverable strain, which is considered to be a representative measure for the molecu-lar orientation. The underlying idea is that, not the flow kinematics, but the behavior of molecules influences the speed of nucleation. Koscher and Fulchiron [211] considered the first normal stress difference as a driving force for nucleation. Coppola et al. [67] modeled the flow-enhanced nucleation based on free energy considerations. They took the right-hand side of Equation 10.18 to construct the function f and modified it by adding an extra term
∆Ff onto∆Fqin the equation, where∆Ff is the flow-induced free energy change, drawn from Doi-Edwards model. Zheng and Kennedy [418] also considered the function f as a function of the free energy, and they used the FENE-P dumbbell model to calculate∆Ff. Both the Doi-Edwards free energy and the FENE-P dumbbell free energy exhibit the following limiting be-havior in shear flows [67, 359]. For low shear rates, one has
∆Ff
n0kBT ∝ (λaγ)˙2, (10.26)
and for high shear rates, one has
∆Ff
n0kBT ∝ ln(λaγ),˙ (10.27)
where n0is the number density of the molecules, kB is the Boltzmann’s constant,λais the relaxation time of the polymer melt, and ˙γ is the generalized strain rate. The quantity n0kBT is normally related to the shear modulus that can be estimated or measured.
Tanner and Qi [359] also proposed a strain and strain-rate formulation for f :
f = a ˙γpγ, (10.28)
where a and p are empirical constants, andγ = ˙γtsis the strain with tsthe shearing time. The equation gives a very good fit to experimental results of Wassner and Maier [396] in simple shearing and small-train oscillation.
10.2.2 Critical Parameters for Shish-Kebab Structure Formation
As mentioned above, the effect of flow can also introduce a morphological change from isotropic spherulites to oriented crystalline shish-kebab structures. Eder and Janeschitz-Kriegl [91] assume that the influence of flow on this type of morphology is due to the formation of thread-like precursor from which lamellae grow perpendicularly.
Experimental observation of Mykhaylyk et al. [263] shows that two parameters are responsible for the formation of shish-kebab morphology by flow-induced crystallization: a critical shear rate ˙γcbelow which the oriented structure is unlikely to be formed, and a critical amount of a specific work, wc, which is required to create oriented nuclei to form shish-kebab structures for shear rates above ˙γc.
The critical shear rate is estimated by ˙γc≈ 1/λR, whereλRis the longest Rouse relaxation time of molecules [81, 312], proportional to the square of molecular weight. The specific work, w , is given by the integral of the product of the viscosity,η, and the square of the strain rate over the total shearing time, ts:
w = Z ts
0 η[ ˙γ(t)] ˙γ2(t )d t . (10.29)
Therefore, the specific work contains the effect of shearing time in addition to the strain rate.
The governing criterion to signify the formation of shish-kebab structure is thus given by
γλ˙ R> 1 and Z ts
0 η[ ˙γ(t)] ˙γ2(t )d t > wc. (10.30)
Mykhaylyk et al. [263] found that the critical specific work is independent of the shear rate, but decreases with increasing concentration of long chains. These observations are further supported by the experimental work of Housmans et al. [161]. Based on these results, Steen-bakkers and Peters [341] and SteenSteen-bakkers [340] modeled the flow-enhanced nucleation rate ( ˙Nf) using a fourth-order dependence on the molecular stretch of the slowest relaxation mode:
N˙f =
gn(Λ4− 1)
· 1 − Nf
Nf ,max
¸
if γ>0˙
0 if γ = 0˙
, (10.31)
where gn is a scaling parameter that depends on temperature only, Nf ,maxis the saturated number density, andΛ is the stretch of the high molecular weight chains that can be deter-mined from the chosen viscoelastic constitutive equation. Based on the experimental evidence
152 10 Crystallization
that a critical specific work needs to be exceeded to change from the isotropic growth regime to the oriented regime, a critical molecular stretchΛcritis also introduced that acts as a threshold to generate oriented crystalline morphologies onceΛ > Λcrit.
Mykhaylyk et al. [263] also showed that the critical shear rate and the critical work can be mea-sured experimentally. The plate-plate geometry can be used to produce a radial distribution of shear rates across a sheared sample (where the shear rate at the radial position r is given by ˙γ = ωr /d where ω is the angular velocity of the rotating plate and d is the gap between the plates). Structure-related methods such as small-angle X-ray scattering (SAXS) and birefrin-gence can be used to identify the boundary between isotropic and oriented structures.
10.2.3 Kinetics Equation of Shish-Kebab Structure
For the purpose of modeling, the geometry of a shish-kebab can be represented by a cylinder.
The fictive volume fraction of the shish-kebabs is then given by
φ = πLtotal
·Z t
0
G(u)d u
¸2
, (10.32)
where Ltotalis the total length of threads per unit volume, which can be calculated by
Ltotal= Z ts
0
N˙f(s)l (ts− s)d s, (10.33)
where ts is the shearing time and l (ts− s) is the length of the thread-like nucleus at time ts, activated at time s. For more details and further references, see Janeschitz-Kriegl [181].
If we denote the fictive volume fractions of the spherulites and the shish-kebabs byφspher es
andφr od s, respectively, we can calculate the relative crystallinity by
α = 1 − exp£−(φspher es+ φr od s)¤ . (10.34)
10.2.4 Material Characterization for Flow-Induced Crystallization Kinetics
To measure crystallinity and the half-crystallization time for flow-induced crystallization, one cannot use conventional DSC, since it does not allow one to impose a flow on the sample.
One may use the Linkam shearing hot stage combined with a microscope and a light intensity reader (Koscher and Fulchiron [211]; Hadinata et al. [142]) . In the experiment, the light in-tensity transmitted through the sample between crossed polarizers is recorded with time. The experiment begins with an initial light intensity I0, and is finished when the intensity reaches a stable value I∞, indicating that crystallization is completed. The relative crystallinityα is estimated as follows:
α(t) =I (t ) − I0
I∞− I0
, (10.35)
and the half-crystallization time is estimated as the time giving the intensity
I1/2(t1/2) =I∞− I0
2 . (10.36)
The variation of morphology through the thickness direction of an injection-molded part can be investigated using synchrotron small-angle and/or wide-angle X-ray techniques. Figure 10.7 shows an example of small-angle X-ray scattering (SAXS) experimental results reported by Zhu and Edward [428] for a sample of iPP. The figure shows 2D SAXS image patterns at differ-ent distances from the wall surface to the mid-surface through the sample thickness. At 100 and 200µ, we can see two distinct maxima, one along the equatorial direction and the other in the meridional direction, indicating the formation of oriented shish-kebab structure. The equatorial and the meridional maximums are attributed to the shish structure parallel to the flow direction and the kebab structure perpendicular to the flow direction, respectively. At 300 µ away from the wall, the equatorial maximum decreases significantly, while the meridional maximum shows little change. At 500µ, both the equatorial and meridional maxima become very weak. With the distance further increasing toward the mid-surface, the image patterns indicate isotropic structures.
Detailed descriptions on characterization of flow-induced crystallization can be found in van Erp [379].
Figure 10.7 Two-dimensional SAXS image patterns at different distances from the skin surface to the mid-surface for an iPP (from Zhu and Edward [428], with permission from American Chemical Society)