kinetics of oriented crystallization of polymers in the ... · fiber velocity, diameter, tensile...

1. Introduction

Molecular deformation and orientation in the amor-phous phase strongly accelerate crystallization ofpolymers. The orientation is caused by tensile stress-es applied during processing. In polymer fluids, mo-lecular orientation produced during melt spinning offibers, film blowing, film casting, etc., is controlledby the deformation rate, or stress. Extension and ori-entation of chain macromolecules in polymer fluidsunder the deformation rate result in enhancement oftensile modulus and tenacity of the material, while theenhanced crystallization kinetics stabilizes polymerstructure and properties. The tensile stresses and mo-lecular orientation affect the dynamics of processingof crystallizing polymers and kinetics of structureformation. Besides high impact on the crystalliza-tion rate, the orientation influences thermodynamic

behavior of the polymer and rises the equilibriummelting temperature.In this work we focus on existing problems in mod-eling of crystallization kinetics under uniaxial mo-lecular orientation, typical for fiber melt spinningprocesses. High values of the elongation rate in meltspinning processes result in significant amorphousorientation of the chain macromolecules. Dynamicsof the processes is strongly coupled with fast crys-tallization induced by the orientation, in particularfor slowly crystallizing polymers. Molecular orien-tation in the amorphous structure and the degree ofcrystallinity determine mechanical and physicalproperties of fibers. The role of orientation in the de-velopment of polymer structure and properties hasbeen a subject of main interest on the fundamentalsof melt spinning processes [1–7].

330

Kinetics of oriented crystallization of polymers in the linear

stress-orientation range in the series expansion approach

L. Jarecki*, R. B. Pecherski

Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland

Received 11 September 2017; accepted in revised form 29 November 2017

Abstract. An analytical formula is derived for the oriented crystallization coefficient governing kinetics of oriented crystal-lization under uniaxial amorphous orientation in the entire temperature range. A series expansion approach is applied to thefree energy of crystallization in the Hoffman-Lauritzen kinetic model of crystallization at accounting for the entropy of ori-entation of the amorphous chains. The series expansion coefficients are calculated for systems of Gaussian chains in linearstress-orientation range. Oriented crystallization rate functions are determined basing on the ‘proportional expansion’ ap-proach proposed by Ziabicki in the steady-state limit. Crystallization kinetics controlled by separate predetermined and spo-radic primary nucleation is considered, as well as the kinetics involving both nucleation mechanisms potentially present inoriented systems. The involvement of sporadic nucleation in the transformation kinetics is predicted to increase with in-creasing amorphous orientation. Example computations illustrate the dependence of the calculated functions on temperatureand amorphous orientation, as well as qualitative agreement of the calculations with experimental results.

Keywords: modeling and simulation, kinetics of oriented crystallization, amorphous orientation, sporadic nucleation, pre-determined nucleation

eXPRESS Polymer Letters Vol.12, No.4 (2018) 330–348Available online at www.expresspolymlett.comhttps://doi.org/10.3144/expresspolymlett.2018.29

*Corresponding author, e-mail: [email protected]© BME-PT

Experimental investigations on oriented crystalliza-tion have shown that the crystallization rate andequilibrium melting temperature increase with theamorphous orientation, and the transformation rateincreases by orders of magnitude [8–15]. It has beenshown that the effects are controlled by the entropyof deformation and orientation of chain macromole-cules in the amorphous component [2, 16–25]. Ori-ented crystallization during melt spinning stronglyinfluences the process dynamic and axial profiles offiber velocity, diameter, tensile stress, birefringence,amorphous and crystalline orientation, degree ofcrystallinity [4, 5, 26, 27]. The increase in crystal-lization rate is estimated for several orders of themagnitude in fast melt spinning processes. Rapid ori-ented crystallization causes hardening of the poly-mer which prevents further drawing of the filamentson the spinning line and introduces limitation in theattenuation of the fiber and achievable spinningspeed [4, 5, 28]. The influence of crystallization onthe rheological behavior has been investigated ex-perimentally, as well as by using modeling concepts[24, 29–40].Significant effects of oriented crystallization hasbeen indicated by modeling of melt spinning process-es that include effects of hardening of the polymerby online crystallization, such as high-speed meltspinning [4, 5, 24, 28], melt blowing [41–43], pneu-matic melt spinning in the Laval nozzle under super-sonic air jet [43, 44].In the modeling of polymer processing under vari-able molecular orientation and temperature condi-tions, the crystallization rate has been expressed bya quasi-static formula [5, 6] obtained by an extensionof the Avrami-Evans formula basing on the non-isothermal Nakamura approach [45, 46]. The Naka-mura approach extended for oriented crystallizationis considered as superior for predicting the develop-ment of crystallization [5]. The instantaneous crys-tallization rate reduces to a simple formula derivedbasing on an orientation- and temperature-dependentcrystallization half-time in the quasi-static approxi-mation, a directly observable kinetic characteristicsof crystallization at fixed orientation and tempera-ture [46, 47]. The half-time analysis has been usedby Spruiell and coworkers [4, 5] to discuss problemswhich appear in applying the experimental kineticsdata in modeling of real melt spinning processes.Other mechanisms possible under variable molecu-lar orientation and temperature, such as athermal

nucleation [48–50] and memory effects [51–54] whichresult from transient distribution of cluster sizes [50,55] and transient molecular orientation [56, 57] havebeen indicated by Ziabicki and coworkers [46, 49,55, 58–60] and Jarecki [61] basing on the Avrami-Evans approach.At large undercoolings, the nucleation mechanismmay no longer be valid and a low temperature limithas been indicated for nucleation-controlled crystal-lization [55]. Then, the crystallization rate cannot bedescribed by the nucleation-controlled mechanism.A failure of the crystallization half-time approachhas been detected in the experimental investigationsat the combination of large undercoolings and stress-es in melt spinning, possibly due to a collapse of thekinetic model of crystal nucleation under such con-ditions [5, 62].A phenomenological formula for the crystallizationrate function has been proposed by Ziabicki [2, 47]basing on the analysis of the crystallization half-timeof isothermal, oriented crystallization at constantmolecular orientation. Series expansions of the ratefunction over the amorphous orientation factor hasbeen considered in the analysis using the kinetic the-ory of nucleation and crystal growth with the effectsof molecular orientation in the transformation freeenergy. In the phenomenological formula, the ratefunction increases exponentially with the square ofthe amorphous orientation factor multiplied by apositive phenomenological coefficient. The orientedcrystallization coefficient has been estimated fromindependent experiments on pre-oriented PET fibersby Smith and Steward [9], Stein [10], Alfonso et al.[11] and its values increase with increasing crystal-lization temperature. Similar estimates of the coef-ficient and its temperature dependence have beenprovided by Spruiell [4, 5, 27] from measurementsof oriented crystallization rate during high-speedmelt spinning of Nylon 6.Several phenomenological formulas are provided forthe temperature dependence of the oriented crystal-lization coefficient in the literature [2, 4, 11] and allof them show inverse dependence on the undercool-ing. The formulas have been determined in the rangeof large undercoolings, and there is no reliable in-formation, experimental or theoretical, from whichthe coefficient could be determined in the entire tem-perature range. Modeling of melt spinning processeswith the oriented crystallization effects indicates thatthe oriented crystallization coefficient is responsible

Jarecki and Pecherski – eXPRESS Polymer Letters Vol.12, No.4 (2018) 330–348

331

for important effects in the process dynamics andstructure formation [4, 5, 28, 41–44]. None of the ac-tually available models are satisfactory for calculat-ing the kinetics of oriented crystallization under vari-able temperature in the entire range of undercoolingwith the use of the oriented crystallization coefficientproposed by Ziabicki [2, 47].Another formula derived basing on the analysis ofthe crystallization half-time and the kinetic theory ofcrystallization [63] has been proposed by Spruiell andcoworkers [7, 27]. A failure of the formula [7, 27] topredict effects of oriented crystallization in the mod-eling of fiber melt spinning may be due to a failureof the nucleation mechanism at combined large un-dercoolings and high orientations. Possibility of sucha failure has been indicated by Ziabicki et al. [55] forlarge undercoolings and by George [62] for processesunder high molecular orientation. It is indicated [55,62] that under the extreme undercooling conditionscombined with high amorphous orientation, the crys-tallization mechanism changes from nucleation-con-trolled one to a homogeneous, diffusion-controlledprocess throughout the melt with zero free energybarrier of nucleation. However, kinetics of orientedcrystallization under such extreme conditions in fastmelt spinning apparently has been adequately de-scribed by the phenomenological formula proposedby Ziabicki, with the use of quiescent kinetic data.Development of amorphous orientation is stronglycoupled with the crystallization in processes underfast elongational flow. Under large tensile stresses,non-linear effects in the development of orientedamorphous structure vs. stress is predicted [25, 56,60, 64, 65]. Closed-form analytical formula for non-linear stress-orientation relation is derived for the en-tire range of tensile stresses, or orientations, basingon the inverse Langevin chain statistics which ac-counts for finite extensibility of the chain macromol-ecule in the amorphous rubbery network [66].The formula for temperature dependence of the ori-ented crystallization coefficient is still needed to pre-dict crystallization rate under large tensile stresses andfast cooling in the entire range of undercoolings. Inthe present paper, an analytical formula for the coef-ficient is derived basing on the kinetic model of crys-tallization [63] and experimental indications [5, 7, 27].

2. Crystallization rate formulations

Progress of crystallization is controlled by nucle-ation followed by crystal growth and its rate shows

a maximum at an intermediate temperature betweenthe equilibrium melting point and the glass transitiontemperature. Under non-isothermal conditions wherethe temperature changes over the range of variablecrystallization rate the transformation progress is af-fected by the cooling rate. At faster cooling the crys-tallization time is shortened at each temperature inter-val and the progress of crystallization is suppressed.Molecular orientation produced under external ori-enting stresses is one of the processing factors thatcontributes to the thermodynamic driving force ofcrystallization and may significantly speed up thetransformation rate. The orientation-induced enhance-ment of the crystallization kinetics applies to the en-tire crystallization temperature range and significantlymodifies the influence of the cooling rate on crystal-lization at any processing. The orientation results inan earlier crystallization and enables crystallizationin the case of slowly crystallizing polymers duringfast cooling. In high-speed melt spinning processesthe rates of online crystallization under molecular ori-entation increase by orders of magnitude and the en-hancement in the transformation kinetics is higher athigher temperatures [2, 3, 7].Under isothermal conditions and zero amorphousorientation, the progress of crystallization is de-scribed by the Avrami equation θ(t) = 1 – exp(–ktn)where θ(t) = X(t)/X∞ is the conversion ratio by crys-tallization, X(t) and X∞ are volume degrees of thetransformation, the instantaneous one and at full con-version, respectively, k the crystallization rate con-stant dependent on the temperature, n the Avrami ex-ponent.The progress of non-isothermal crystallization is de-scribed by the Nakamura equation [45] θ(t) =1 – exp[–(

0∫tK(T)dt′)n] at an isokinetic approximation

where the temperature-dependent crystallization ratefunction K(T) = [k(T)]1/n. Under quiescent condi-tions, the rate function is described by the phenom-enological formula [2]:K(T) = Kmaxexp ⎣–4ln2(T – Tmax)2/D2⎦.

In our approach we follow the classical line similarin the form to the Kolmogoroff-Avrami-Evans the-ory in describing the crystallization kinetics, devel-oped for the transformation under variable tempera-ture and molecular orientation [45, 46]. We accountfor the effects of variable amorphous orientation byderiving an analytical formula for the crystallizationrate function K(T,fa) basing on the ‘proportional ex-pansion’ approach proposed by Ziabicki [46] for the


332

crystallization kinetics under variable temperatureand orientation.We confine our considerations to constant mecha-nisms in the crystallization kinetics governed bythermal nucleation, sporadic and predetermined, andnucleation-controlled crystal growth in the quasi-sta-tic approximation. Athermal and memory effects arenot accounted for. The quasi-static crystallizationrate under variable temperature and orientation is ex-pressed by the formula obtained by an extension ofthe Nakamura et al. [45] approach for oriented crys-tallization [5, 6] (Equation (1)):

(1)

where K is the crystallization rate function depend-ent on the actual temperature T(t) and the actual ori-entation factor fa(t) . Equation (1) is equivalent to theformula obtained in the steady-state limit of a moregeneral ‘proportional expansion’ approach [46] andis considered as a superior one for predicting thetransformation rate in processes under variable tem-perature and orientation.Equation (1) in the integral form reads (Equa-tion (2)):

(2)

where the kinetics and the degree of the transforma-tion are determined by the temperature – and orien-tation – dependent rate function K and its history.Let us observe that in this discussion an analogy ap-pears with the concept of the development of themicro-shear bands contribution in modeling of thedeformation mechanisms in polymers and metals[67, 68].The rate function in Equations (1) and (2) expressesby the inverse of the observable crystallization half-time, K(T,fa) = (ln2)1/n/t1/2(T,fa), determined for qui-escent crystallization at fixed temperature T and ori-entation fa [46, 47]. Constant Avrami exponent n andconstant mechanisms of nucleation-controlled crys-tallization are assumed. The rate function has beenapproximated by a phenomenological formula bas-ing on the analysis of the crystallization half-time ofisothermal crystallization at constant orientation byZiabicki [2, 47]. In the analysis, series expansions ofthe rate function over the factor fa has been consid-ered in the linear stress-orientation range basing onthe kinetic theory of nucleation and crystal growth

with the contribution of orientation to the crystalliza-tion free energy. The series expansion has been con-fined to the square term in fa [2, 47] (Equation (3)):

(3)

where the term linear in fa had dropped out for sym-metry reasons and the crystallization rate increasesexponentially with A(T)fa2. The global rate functionhas been considered in the series expansion approachand Equation (3) has been derived without consid-ering the contributions of sporadic and predeter-mined nucleation, as well as crystal growth.There is no reliable information from which the tem-perature dependence of the oriented crystallizationcoefficient A could be determined for various poly-mers, except for some rough estimates from experi-mental investigations for a few polymers at large un-dercoolings. The magnitude of A has been estimatedto be in the range 102–104 from independent exper-iments on the crystallization rate of pre-oriented PETsamples, with the values increasing at increasing tem-perature [9, 11]. Large values of A indicate strongeffects of amorphous orientation in the crystalliza-tion kinetics in real processes.Large values of A and their increase with tempera-ture have been shown by Spruiell et al. [4, 5, 27] byonline measurements of the crystallization rate dur-ing melt spinning of nylons. Although the phenom-enological formula, Equation (3), has been derivedbasing on the series expansion in the linear stress-orientation range, experimental investigations indi-cate validity of the formula also above the linearrange, fa > 0.1 . In the present work we derive aclosed-form formula for the coefficient A(T) basingon fundamental formulations of the kinetic theory ofcrystal nucleation and growth [63]. The influence ofsporadic and predetermined nucleation on A(T) is ac-counted for.The experimental information on the material func-tion A(T) still remains scarce, at the lack of any for-mula derived from the first principles. Large difficul-ties are associated with experiments designed todetermine the rates of oriented crystallization at con-trolled amorphous orientation and temperature. Oneconcludes from modeling of fast melt spinningprocesses that the function A(T) is responsible for im-portant effects in the processing dynamics and struc-ture formation. For example, online solidification of

, lnddt nK T t f t 1 1

/a

n n1i i i= - -- -Q Q Q Q QV V V V V" "% %

,exp dt K T s f s s1 a

t n

0

i = - -Q R Q QV V VW# &G J#

, exp lnK T f KD

T TA T f4 2max

maxa a2

2

2= --

+R R QW W V# &


333

the polymer by crystallization is predicted for slowlycrystallizing polymers which significantly limits at-tenuation of the spun melt and the take-up velocity.Another formulation of the rate function K(T,fa) pro-posed by Patel and coworkers [7, 27] bases on theHoffman kinetic theory of crystallization [63] limit-ed to the contribution of predetermined nucleationfollowed by crystal growth (Equation (4)):

(4)

where K(T,0) is the rate function at temperature Tand fa = 0, Tm

0 is the equilibrium melting temperature,C3 material constant determined from isothermal crys-tallization rate, C material constant estimated fromonline measurements of the crystallization rate.An analysis of available approaches in modeling ofmelt spinning [5] has indicated that Equation (4)could not even qualitatively predict oriented crystal-lization rate at high spinning speeds, while it has pre-dicted the rate at lower spinning speeds where nocrystallization had been observed. It is claimed [5]that the failure of Equation (4) may result from thelack of nucleation mechanism in crystallization undervery high orientation combined with large undercool-ings. One has concluded in the analysis however, thatthe crystallization kinetics has been adequately de-scribed by Equation (3) with the phenomenologicalfactor exp(Afa2) also under the extreme conditions.

3. Crystallization rate under variable

orientation and temperature

We calculate the rate functions for crystallizationunder variable temperature and orientation basing onthe Hoffman-Lauritzen kinetic theory [63] account-ing for sporadic and predetermined primary nucle-ation, as well as for crystal growth. Basing on the ki-netic theory and the experimental indications [5, 7,27], analytical formulas for oriented crystallizationcoefficient A(T) are derived in this paper. First, for ori-ented crystallization involving separate sporadic andpredetermined nucleation, and next for processeswith both types of primary nucleation present in thesystem. Our calculation are based on the ‘propor-tional expansion’ approach proposed by Ziabicki[46] in the steady-state limit.The function K(T,fa) for process under variable tem-perature T(t) and orientation fa(t) is inversely propor-tional to the instantaneous crystallization half-time(Equation (5)):

(5)

For processes with isolated sporadic nucleation andfor processes with predetermined nucleation (hetero-geneous, memory nuclei) the rate functions read, asshown in Equations (6) and (7) respectively [46]:

(6)

(7)

where N·(T,fa) is the steady-state homogeneous nu-cleation rate, N0 the number of predetermined nucleiper unit volume, G(T,fa) linear growth rate at T andfa, n Avrami exponent, R0 dimension of the crystallitein which the growth does not proceed.From the kinetic model of Hoffman-Lauritzen, thesporadic nucleation and crystal growth rates read asshown in Equations (8) and (9):

(8)

(9)

where ∆g(T,fa) is free energy of crystallization perunit volume of the crystalline phase at T and fa, Ns

the number of single kinetic units per unit volumeof the amorphous phase identified with statisticalsegments of the chains, kT/h thermal frequency, U*

activation energy of molecular transport, Tm temper-ature at which molecular motions cease near thecrystal surface, σe and σ free energy densities at theend and side surfaces, b thickness of the surface nu-cleus, c constant dependent on the nucleus geometry,h and R are the Planck and gas constants. The pre-exponential factor G0 is proportional to thermal fre-quency.The tensile stresses determine strain energy and con-tribute to the enthalpy and entropy of the amorphousand crystalline phases. The thermodynamic func-tions of the phases modified by the orientation fa attemperature T read:– for the enthalpy and entropy of the amorphous

phase in Equations (10a) and (10b):

(10a)

, . expK T f K TT T T CT f

C0a

m a0 2 2

3=

- +R Q RW V W# &

,,

lnK T f

t T f2

/

/

aa

n

1 2

1

=R QRW V

W

, , ,K T fn N

T f R G T f1/

a an n

a

n

04 1

1

= - -oR R RW W W# &

, ,K T f N R G T f/

a

n

an

0 03

1

= -R R RW W W

,,

expN T f N hkT

R T TU

kT g T f

c2*

a sa

e2

2 2v v

D= -

--

3

o R R RW W W" %G J

,,

expG T f GR T T

U

kT g T f

b4*

aa

e0

vv

D= -

-+

3

R R RW W WG J

, ,h T f h T h T fa a a a a

0 d= +R Q RW V W


334

(10b)

– and for the crystalline phase in Equations (11a) and(11b):

(11a)

(11b)

where δha, δsa, δhc, δsc are the contribution of theorientation to the thermodynamic functions at tem-perature T and ha

0, sa0, hc

0, sc0 are the functions at zero

orientation and T.In general, the contributions of orientation to thethermodynamic functions modify the crystallizationfree energy density (Equation (12)):

(12)

where ∆g0(T) = ∆h0(T) – T∆s0(T) and ∆h0, ∆s0 arethe enthalpy and entropy of crystallization in the un-oriented state. The entropy of amorphous phase de-creases faster than the entropy of the crystallinephase with decreasing T. The decrease in the en-tropies at decreasing temperature is accounted for inthe entropy of crystallization of the unoriented poly-mer by a reducing factor f(T) ~= 2T/(T + Tm

0), the sameas for the decrease in the crystallization enthalpy,and reads ∆s0(T) = f(T)∆hm/Tm

0 where Tm0 and ∆hm are

the equilibrium melting point and the melting heat,respectively. With the reducing factor the crystalliza-tion free energy in the unoriented state reads∆g0(T) = f(T)∆hm∆T/Tm

0 where ∆T = Tm0 – T is the un-

dercooling [63].In oriented crystallization, the largest contribution oforienting stresses to the crystallization free energy∆g(T,fa) concerns entropy of orientation of the amor-phous phase, δsa, while the remaining contributionsδsc, δha and δhc are considered as relatively smalland usually are neglected [25, 69, 70]. We neglectthe temperature effects in the entropy contributionδsa, in comparison with large contribution of the con-figurational entropy of the chains under orientingstresses, and the contribution is approximated by thechange in configurational entropy under orientation,δsa(T,fa) ~= δsa

conf(fa) < 0. The approximation is con-firmed by entropic elasticity of the flexible chainamorphous polymers determined by configurationalentropy of the chains.

With the orientation effects in the crystallizationdriving force approximated by the change in config-urational entropy of amorphous chains, free energydensity of crystallization is enhanced by amorphousorientation and reads as shown in Equation (13):

(13)

The orientation entropy contributes negative term tothe transformation free energy and considerablyspeeds up the crystallization. The equilibrium melt-ing point also increases with increasing orientationaccording to Equation (14):

(14)

where δsaconf(fa)/∆s0(T) > 0. The increase in the equi-

librium melting point under orientation results inan increase of the undercooling by aboutTm

0 δsaconf(fa)/∆s0(T), without changing the tempera-

ture. With the thermodynamic effect of orientation,the nucleation rates of sporadic nucleation and crys-tal growth increase exponentially due to reductionof the thermodynamic barriers of nucleation, Equa-tions (8) and (9).A more detailed analysis of the other contributionsof orienting stresses to the crystallization free energyincluding enthalpy of the amorphous chains and thecrystals strain energy shows that all these effects,neglected in the present approach, contribute to high-er undercooling, higher crystallization driving forceand faster nucleation rates [23, 25, 69, 70].

4. The series expansion approach

The change in the configurational entropy of theamorphous chains under the orientation is consid-ered as sufficiently smooth function of the Hermansaxial orientation factor fa and is expanded in thepower series over fa around zero orientation, fa = 0.The series expansion reads as shown by Equa-tion (15) [2, 47]:

(15)

where the zero-order and linear terms drops out be-cause of physical and geometrical reasons [2, 47],i.e. the entropy of the amorphous phase in orientedstate cannot exceed entropy of the unoriented phaseat positive or negative fa. With the series expansion,the rates of sporadic nucleation and crystal growthread as shown by Equation (16) and (17):

, ,s T f s T s T fa a a a a

0 d= +R Q RW V W

, ,h T f h T h T fc a c c a

0 d= +R Q RW V W

, ,s T f s T s T fc a c c a

0 d= +R Q RW V W

, , ,

, ,

g T f g T h T f h T f

T s T f s T f

a c a a a

c a a a

0 d d

d d

D D= + - -

- -

R QR

RR

RW VW

WW

W" %

,g T f g T T s fa aconf

a0 dD D= +R Q RW V W

T f

s T

s f

TT

s T

s f

1

1m aa

mm

a

0

00

0aconf

aconf

,d

d

D

D=

-

+RQR Q

RWVW V

W# &

s f k a f a fa a a22

33

aconf

fd =- + +R RW W


335

with the lowest term quadratic in fa in the exponentialfunctions responsible for the orientation effects.In the series expansion approach, the rate functionof oriented crystallization controlled solely by spo-radic nucleation reads as shown by Equation (18):

(18)

where the temperature-dependent oriented crystalliza-tion coefficients are expressed by Equations (19a)and (19b):

(19a)

(19b)

For process controlled solely by predetermined nu-cleation we have Equations (20) and (21):

(20)

(21a)

(21b)

The pre-exponential factors Ks0(T) and Kp

0(T) in Equa-tions (18) and (20) are the rate functions at fa = 0.

5. The linear stress-orientation conditions

Extension and orientation of chains along the tensilestress axis affects thermodynamics and kinetics ofcrystallization. Due to exponential stretch relaxationpresent in polymeric fluids, orientational effectivityof uniaxial elongational flow is determined by theproduct of the flow elongation rate and the stretchrelaxation time, q·τ, while the effectivity of the pro-cessing time is controlled by the reduced time, t/τ.The same behavior is shown by the configurationalentropy of the chains and the elastic tensile stresses

in such systems [2, 25, 60, 66]. Modeling of time-evolution of molecular deformation of the chains inpolymer melts under uniaxial elongational flow isbased on the evolution equations (Equations (22a)and (22b)) for the average chain extension coeffi-cients λ and λ⊥ along and perpendicular to the flowaxis [60, 66]:

(22a)

(22b)

where E(t) is the Peterlin non-linear modulus of elas-ticity which varies in time between unity in the linearGaussian range and infinity at full chain extension.The amorphous orientation factor fa and the averagetensile stresses ∆p have been calculated in [60, 66]basing on a non-linear inverse-Langevin chain sta-tistics in systems subjected to uniaxial elongationalflow with the molecular stretch relaxation accountedfor. The calculated stress-orientation relations coincideinto a unique, single master plot in the entire rangeof the elongation rates, independent of the elongationrate and the relaxation time because the tensile stressand orientation undergo the stretch relaxation simul-taneously. The unique stress-orientation relations be-havior is a consequence of the fact that the molecularorientation is controlled solely by the stress.The stress-orientation formula shows linearity in theGaussian range of chain extensions which produceorientation in the range of fa between zero and ofabout 0.1. In the linear range, the average molecularelongation coefficients, with the stretch relaxationaccounted for, follow the relation λ(t)λ⊥

2(t) ~= 1 duringthe processing time [60, 66]. The average tensilestress in the linear range ⟨∆p⟩(t) = νkT⎣λ2(t) –1/λ(t)⎦and the orientation factor fa(t) = ⎣λ2(t) –1/λ(t)⎦/(5N)where ν is the number of elastic chains per unit vol-ume, N the number of statistical segments per chain.At higher tensile stresses, or orientations, present forexample in high-speed melt spinning of fibers, non-linear inverse Langevin stress-orientation behavioralso does not show up explicitly the molecular stretch

A Tf h

b

T

T

n bf hc

T

Tn

n a4 1 1

s

m

e m

m

m2

0 22 0

2

vv vD D D D

= +-Q

RU UV

WZ Z

B T aa

A Ts s2

3=Q QV V

, expK T f K T A T f B T fp a p p a p a0 2 3

f= + +R Q Q QW V V V" %

B T aa

A Tp p2

3=Q QV V

A Tf h

b

T

Ta

4p

m

e m2

0 2

2

vv

D D=Q

RUV

WZ

, expK T f K T A T f B T fs a s s a s a0 2 3

f= + +R Q Q QW V V V" %/d

dt

E t q2 1 02

2

xm x m+ - - =oQ QV V" %

/d

d

tE t q 1 0

22

xm

x m+ + - ===

oQ QV V" %


336

(16)

(17), , expG T f G T ff h

b

T

Ta f a f0

4a a

m

e ma a2

0 2

22

33

fvv

D D= = + +R R R U RW W W Z W# &

, , expN T f N T ff h

c

T

Ta f a f0

4a a

m

e ma a3

2 2 0 3

22

33

fv v

D D= = + +o oR R R U RW W W Z W# &

relaxation which is hidden in the master stress-ori-entation plot [60, 66].In drawing deformation of solid elastic amorphouspolymers, molecular deformation and orientation arecontrolled by the draw ratio, not by the elongationrate or stress, and the contribution of the molecularstretch relaxation is strongly limited [2, 65]. Cross-linked solids in rubbery state also show the linearGaussian range in the stress-orientation behavior andan abrupt increase in the stress due to a limited ex-tensibility of the network chains above the linearrange, similar to the master plot determined for poly-mer melts. The difference with the flow deformationof polymeric melts is that the molecular elongationcoefficients in the rubbery solids coincide with themacroscopic elongation coefficients. For isochoricuniaxial drawing of a rubbery cross-linked solid therelation λλ⊥

2 = 1 also holds in the linear stress-orien-tation range [71].We determine the series expansion coefficients inEquation (15) for systems of flexible linear chainssubjected to uniaxial tensile stresses in the linearstress-orientation range under uniaxial elongationalflow or uniaxial elongation of rubbery solids. Theaverage entropy of deformation in the Gaussianrange per unit volume of the amorphous phase ap-proximated by the change in configurational en-tropy of the chains [8, 19, 60] δsa

conf = ν⟨δSaconf⟩ =

– νk(λ2 + 2/λ – 3)/2 where ⟨δSaconf⟩ is an average

decrease in the configurational entropy per singlechain.The series expansions of the deformation entropyand the orientation factor over the powers of λ – 1read as shown by Equations (23) and (24):

(23)

(24)

where the zero- and first-order terms in the series ex-pansion of δsa

conf and the zero- and second-order termsin the expansion of fa drop out.The inverse series expansion applied for Equa- tion (24) reads (Equation (25)):

(25)

where the second-order term drops out. With Equa-tion (25), the decrease in the configurational entropyper unit volume caused by the orienting stresses isexpressed by the series expansion over fa and isgiven by Equation (26):

(26)

The number of statistical segments in the linear chainundergoing elastic deformation is N = nbM/(C∞m)where M is the chain molecular weight, nb numberof rigid bonds in the chemical unit of the chain, m mo-lecular weight of the chemical unit, C∞ the character-istic ratio. We assume that the elastic chains subjectedto the deformation are identified with the chains be-tween the entanglements with the entanglement mo-lecular weight M = Me. Then, the number of elasticchains per a volume transforming to a unit volumeof the crystalline phase ν = ρcNA/Me where ρc is den-sity of the crystalline phase, NA the Avogadro num-ber and the coefficients are given by Equation (27):

, (27)

6. Discussion

With the aim of deriving the first oriented crystalliza-tion coefficient A(T), we consider the kinetics of ori-ented crystallization in real systems as a process in-volving independent sporadic and predeterminednucleation followed by crystal growth. First, we derivethe coefficients for processes with separate nucleationmechanisms, sporadic and predetermined, and next forprocesses involving both types of nucleation.

6.1. Oriented crystallization coefficients at

separated sporadic and predetermined

nucleation

With the expansion coefficients a2, a3 given by Equa-tion (27), the coefficients for processes with separat-ed sporadic nucleation read as shown by Equa- tions (28a) and (28b):

f N51

3 1 1a3

fm m= - + - +Q QV V" %s

k2 3 1 2 1

2 3aconf

fd om m=- - - - +Q QV V" %

Nf

Nf1 3

531

35

a a

33

fm = + - +S X

sk N

fNf

k a f a f

23

35

910

a a

a a

22 3

22

33

aconf

f

f

d o=- - + =

=- + +RS SX

WX

aC m

N n M

625 c A b e

2 2 2

2t=

3

a C m

n Ma9

10 b e3 2=-

3


337

(28a)A Tf h

b

T

T

n bf hc

T

Tn

nC m

N n M

3

50 1 1s

m

e m

m

m c A b e2

0 22 0

2 2

2vv v t

D D D D= +

-

3

Q R U UV W Z Z

(28b)

and for processes with separated predetermined nu-cleation by Equations (29a) and (29b):

(29a)

(29b)

As an example, we compute the coefficient Ap(T) vs.temperature for processes with predetermined nucle-ation and the ratio As(T)/Ap(T) of the coefficients forsporadic and predetermined nucleation for isotacticpolypropylene (iPP), polyethylene terephthalate(PET), Nylon 66 and poly(L-lactide) (PLLA) usingmaterial parameters listed in Table 1. Large positivevalues of Ap are predicted in the computations. Fig-ure 1 shows Ap plotted vs. undercooling ∆T = Tm

0 – T.Values of Ap increase from of about several hundredat large ∆T to several thousand at decreasing ∆T to30 K in the case of PET, Nylon 66 and PLLA. ForiPP the coefficient is higher by a factor of about twoin the entire range of ∆T. All plots show steep in-crease of Ap tending to infinity at approaching theequilibrium melting temperature Tm

0. Larger valuesof Ap for iPP result from low melting heat at whichcontribution of the configurational entropy of thechain deformation to the crystallization free energyis higher, relative to the crystallization enthalpy. TheAvrami exponent does not influence Ap.Experimental points in Figure 1 show the values ofthe oriented crystallization coefficient A determinedfor PET at large undercoolings [11]. There is lack

of experimental data at lower undercoolings due todifficulties in conducting experiments on orientedcrystallization at very high rates under lower under-coolings. The values of the kinetic coefficient Ashown by the points in Figure 1 have been deter-mined in [11] for PET preoriented amorphous fibersobtained by melt spinning at several take-up speeds.The oriented crystallization rate was measured forsamples characterized by several values of the amor-phous orientation factor, crystallized isothermally atseveral temperatures well below the temperature ofmaximum crystallization rate and quenched after anappropriate time below the glass transition tempera-ture Tg. In determining the orientation factor, formbirefringence was neglected on the assumption thatno voids are present in the fibers melt-spun at lowand moderately high speeds [11]. Crystallization ratewas determined for samples tightly wound on ametal frame to avoid relaxation of the amorphousorientation. Also low temperature range chosen forthe isothermal crystallization, terminated by quench-ing which ensured slow enough non-isothermal crys-tallization at the quenching, indicates high enoughcredibility of the kinetic data. Higher rates of orientedcrystallization at higher temperatures unable ade-quate control of crystallization temperature, orienta-tion and the transformation rate for obtaining credi-ble kinetic data.The experimental points of the oriented crystalliza-tion coefficient A in Figure 1 are of the order of thecoefficient Ap (several hundreds) calculated for PET

B TC m

n MA T9

10p

b ep=-

3Q QV V

B TC m

n MA T9

10s

b es=-

3Q QV V

A Tf h

b

T

T

C m

N n M

3

50p

m

e m c A b e2

0 2

2 2

2vv t

D D=

3

Q R UV W Z


338

Table 1. Material parameters used in the computations

Parameter iPP PET Nylon 66 PLLA

Tm0 [K] 453 553 537 448

∆hm [J/g] 165.0 123.7 188.3 93.1

ρ0 [g/cm3] 1.145 1.356 1.122 0.776

ρ1 [g/(cm3·K)] 9.03·10–4 5.00·10–4 5.45·10–4 4.62·10–4

ρc [g/cm3] 0.938 1.457 1.24 1.37

σ [J/cm2] 1.15·10–6 1.09·10–6 1.07·10–6 1.20·10–6

σe [J/cm2] 6.23·10–6 9.09·10–6 11.8·10–6 6.26·10–6

b [cm] 6.26·10–8 5.95·10–8 3.7·10–8 5.17·10–8

m [g/mole] 42 192 226 72

nb 2 6 14 3

C∞ 5.7 4.0 5.9 10.5

Me 6900 1630 2010 8000

Figure 1. Oriented crystallization coefficient Ap for process-es with separate predetermined nucleation vs. un-dercooling ∆T = Tm

0 – T computed for iPP, PET,Nylon 66 and PLLA from Equation (29a). Exper-imental points for PET obtained on isothermalcrystallization of as-spun amorphous PET fiberswith various amorphous orientation [11].

from Equations (29a) for crystallization controlledby predetermined nucleation and also increase withdecreasing the undercooling ∆T. In the range of ratherlow values of the factor fa of the experimental sam-ples we assign higher credibility to the coefficientAp plotted in Figure 1, rather than to plotting the co-efficient As for sporadic nucleation. Higher experi-mental values of A than the calculated values in Fig-ure 1 for Ap at lower undercoolings ∆T may resultfrom a contribution of an additional crystallizationpossible at quenching from higher temperatures, ac-counted as the isothermal crystallization.Figure 2 shows the plots of the ratio As/Ap of the ori-ented crystallization coefficients computed for iPP,PET, Nylon 66 and PLLA from Equations (28a) and(29a) for processes with sporadic and predeterminednucleation as functions of undercooling ∆T at Avra-mi exponent n = 2. The plots show that the orientedcrystallization coefficient for sporadic primary nu-cleation is higher by a factor of about 4 up to 7 at larg-er undercoolings. At decreasing the undercooling,the ratio of the coefficients increases to higher valuesand tends to infinity at approaching Tm

0. Larger val-ues of As are a consequence of higher impact of theorientation entropy on the free energy barrier of spo-radic nucleation than on the barrier of crystal growth.The influence of orientation on the sporadic nucle-ation free energy barrier and, in consequence, on thecoefficient As is predicted to be lower at higher Avra-mi exponents n. Figure 3 illustrate low influence ofn at large undercoolings at which the As/Ap ratio de-creases by a factor of about 1–2 at increasing n from2 to 4. The decrease is larger at low undercoolings.

No significant difference is predicted for the testedpolymers.

6.2. Rate functions at separate nucleation

mechanisms

Figures 4 show the plots of reduced rate functionKp(T,fa)/Kp

0(T) computed from Equation (20) and(29a, b) vs. amorphous orientation fa2 for processeswith separated predetermined nucleation for PET,Nylon 66 and PLLA at constant undercoolings and


339

Figure 2. The ratio of the oriented crystallization coeffi-cients As/Ap for processes with separated sporadicand predetermined nucleation vs. undercooling∆T = Tm

0 – T computed from Equations (28a) and(29a) for iPP, PET, Nylon 66 and PLLA at n = 2

Figure 3. Ratio As/Ap of the oriented crystallization coeffi-cients vs. undercooling ∆T = Tm

0 – T computed fromEquations (28a) and (29a) for PET (a), Nylon 66(b) and PLLA (c) at n = 2, 3, 4

n = 2. The undercoolings correspond to the temper-ature of the maximum crystallization rate Tmax (in-dicated by the star sign) and in the vicinity of Tmax.Solid line plots illustrate influence of Ap(T) and showhigher effects of the orientation at lower undercool-ings. The dashed lines plots show the influence ofBp(T) which significantly reduces the influence ofAp(T) at increasing fa.

Maximum of the dashed-line plots in Figures 4 is aconsequence of poor convergence of the series ex-pansion. The convergence is poor at higher numberof statistical segments in the chain between the en-tanglements approaching N ≈1/fa where N =nbMe/(C∞m). For the tested polymers, the number Nof statistical segments in the elastic chain is the low-est for PET, increases respectively for Nylon 66,PLLA, iPP, and the convergence becomes poorer inthis order. As reported from melt spinning experi-ments, the phenomenological formula limited to thefirst series expansion coefficient, Equation (3), ade-quately describes kinetics of oriented crystallizationat low, as well as at high fa values, also beyond thelinear stress-orientation range, fa > 1. We focus in thispaper on the first expansion term, proportional to fa2,with the aim to provide analytical formula for the co-efficient A(T) in the phenomenological Equation (3).Our analysis of the crystallization kinetics for process-es with separated nucleation mechanisms allows toestimate the kinetic effects of amorphous orientationin real systems where sporadic and predeterminednucleation may coexist. Much higher values of As(T)calculated for sporadic nucleation in comparisonwith Ap(T). (Figures 3) indicate stronger effects oforientation fa in the transformation kinetics at zerocontent of the predetermined nuclei. Such a case canbe expected for polymers obtained in biosynthesis.In real systems usually both nucleation mechanismare involved and relatively high values of As indicatethat sporadic nucleation may dominate the predeter-mined nucleation at higher orientations and shouldbe considered in the overall transformation kinetics.Usually, predetermined nucleation significantly dom-inates sporadic nucleation under quiescent, zero-ori-entation conditions. Below we estimate the conditionsat which involvement of sporadic nucleation is pro-moted by the orientation to the level of predeter-mined nucleation, and above.

6.3. Rate functions at coexisting sporadic and

predetermined nucleation

In real systems, sporadic nucleation contributes to alower or higher extent and is accounted to as accom-panying predetermined nucleation (heterogeneous,memory nuclei) when considering kinetics of orientedcrystallization in our approach. The presented resultsindicate that above a certain fa value contribution ofsporadic nucleation can dominate predeterminednucleation in the overall crystallization rate. Under


340

Figure 4. Reduced crystallization rate functionsKp(T,fa)/Kp

0(T) vs. fa2 computed from Equation (20)and (29a, b) for processes with predetermined nu-cleation for PET (a), Nylon 66 (b) and PLLA (c)at ∆T corresponding to Tmax (indicated by *) andin its vicinity. Solid lines – influence of Ap(T),dashed lines – influence of Ap(T) and Bp(T).

unoriented conditions, predetermined nucleation isusually the main nucleation mechanism with a neg-ligible contribution of sporadic nucleation. At low ori-entation, predetermined nucleation (if present) canbe still considered as the main nucleation mechanisminvolved in the transformation, while sporadic nucle-ation contributes increasingly with increasing fa.The predetermined and sporadic nucleations are in-dependent processes in real systems. Inverse of thecrystallization half time 1/t1/2 represents a frequencyof achieving 1/2 of the maximum degree of crystallini-ty. Predetermined and sporadic nucleation contributeto the global frequency the frequencies (1/t1/2)p and(1/t1/2)s represented by the rate function of separatesporadic and predetermined nucleation, Kp(T,fa),Ks(T,fa).The contribution of sporadic nucleation in the globalcrystallization mechanism is characterized by theAvrami exponent n, while the component assignedto predetermined nucleation – by n – 1. The role ofsporadic nucleation mechanism at temperature T andorientation fa is characterized by the ratio of the ratefunctions as shown in Equation (30):

(30)

where the exponential function characterizes the in-fluence of sporadic nucleation relative to the influ-ence of predetermined nucleation at T and fa. Thepre-exponential function φ(T) characterizes the in-fluence of sporadic nucleation relative to predeter-mined one at T and zero orientation (Equation (31)):

(31)

where 2 ≤ n ≤ 4. For one-dimensional crystal growthat sporadic or predetermined nucleation we haven = 2 in Equation (31). At n = 4 we have 3-dimen-sional growth at sporadic or predetermined nucle-ation. N0 is the volume density of predetermined nu-clei, G0 the pre-exponential factor of the crystalgrowth rate. In our computations we assume G0 ≈105 cm/s [75].Kinetic units involved in sporadic nucleation andcrystal growth are identified with the chain statistical

segments. Number of the kinetic units per unit vol-ume of the amorphous phase is Ns = ρaNAnb/(C∞m)where ρa is the amorphous phase density. For PET wehave ρa(T) = ρ0 – ρ1(T – 273 K) and for iPP, Nylon 66and PLLA ρa(T) = ρ0/[1 + ρ1(T – 273 K)] where ρ0,ρ1 coefficients are listed in Table 1. We assume thatthe dimension R0 at which the crystal growth doesnot proceed corresponds to the critical dimension ofprimary nucleus, given by Equation (32):

(32)

Example computations of the ratio of the pre-expo-nential function φ vs. ∆T are presented for PET in Fig-ures 5 and 6. In the computations we assume R0 = R*,5R*, 10R* and the predetermined nuclei density N0 =106, 107, 108 cm–3.

,

,exp

K T f

K T fT A T A T f

B T B T f

p a

s as p a

s p a

2

3f

z= - +

+ - +

RR Q

Q QQ QW

W VV V

V V"

"%

%FI

( )

exp

TK T

K T

nN R hG

kTN

kT f h

c

T

TkTf h

b

T

T

1

2 4

/ /

/

p

sn n n

s

m

e m

m

e m

n

n0

0

01

04 1

0

2

2 2 0 2 01

$

$

z

v v vv

D D D D

= =

- -

- - -

QQ

R U

Q Q QVV

W Z

V V V

# &G

J

R V f h

c

T

T2* */

/

m

e m1 32 2

1 30v v

D D= =R RW W


341

Figure 5. Ratio of the crystallization rate functions at zeroorientation φ vs. ∆T computed from Equation (31)for PET at predetermined nuclei density N0 = 106,107, 108 cm–3, R0 = R*, G0 = 105 cm/s, n = 2

Figure 6. Ratio of the crystallization rate functions at zeroorientation φ vs. ∆T computed from Equation (31)for PET at R0 = R*, 5R*, 10R*, N0 = 107 cm–3, G0 =105 cm/s, n = 2

Figures 5 and 6 indicate that the thermodynamic fac-tors associated with undercooling strongly reducethe involvement of sporadic nucleation by orders ofmagnitude, in particular at low undercoolings. Fig-ure 5 shows the reduction by about one order of themagnitude at increasing the content of predeterminednuclei N0 by one order in the range 106–108 cm–3.Rather marginal influence of the crystallite dimen-sion R0 at which the growth does not proceed is il-lustrated in Figure 6. An increase of R0 in the rangebetween the critical value R* and 10R* reduces the in-volvement of sporadic nucleation by about one orderof the magnitude.Very low values of the pre-exponential function φ ina wide range of undercoolings are responsible forstrong reduction of the involvement of sporadic nu-cleation in the global transformation kinetics. Butthe exponential function in Equation (30) is respon-sible for an increase of the involvement of sporadicnucleation with increasing fa because As(T)/Ap(T) > 1(Figures 3) and Bs(T)/Bp(T) > 1. At small fa values, theratio Ks(T, fa)/Kp(T, fa) << 1 due to very low values ofφ(T) and the global rate function is controlled bypredetermined nucleation, K(T, fa) ~= Kp(T, fa).Line plots in Figure 7 show the global rate functionapproximated by the rate function for the processcontrolled by predetermined nucleation K(T, fa) ~=Kp(T, fa) = K0(T)exp⎣Ap(T)fa

2⎦ vs. 103fa calculatedfrom Equation (29a) for PET at ∆T = 165 and 185 Kwith the assumption A(T) = Ap(T) in the range of smallfa values. The experimental points correspond to sev-eral undercooling values in the range 165–185 K[11]. The computed line plots and the experimental

points correspond to large undercoolings and remainwithin the same order of magnitude at low orienta-tions and increase with increasing fa and with de-creasing ∆T. At the highest fa value, the experimentalpoints depart more from the linear plots and indicatesubstantial non-linear effects in the oriented crystal-lization kinetics. The effects will be discussed in therange of non-linear stress-orientation behavior athigher tensile stresses and fa in a separate work.One can expect that with increasing fa, the ratioKs(T, fa)/Kp(T, fa) approaches unity at a certain orien-tation and the involvement of both nucleation mech-anisms in the overall crystallization kinetics is of thesame order of magnitude. Further increase of fa shouldlead to domination of sporadic nucleation in the trans-formation kinetics. Crystallization half-time analysisof oriented crystallization in melt spinning of fibersmade by Patel and coworkers [5, 27] indicates thatin the range of high fa, the orientation effects are alsosatisfactorily described by the function exp⎣A(T)fa2⎦with single phenomenological parameter A(T), thesame as in the range of small fa, proposed by Ziabicki[2, 47].Basing on the series expansion approach, we formu-late a hypothesis that the involvement of sporadicnucleation which is predicted here to increase withincreasing fa is responsible for an apparent agree-ment of the rate function expressed by the factorexp⎣A(T)fa

2⎦ with the experimental observations athigh orientation produced at high-speed melt spin-ning processes [5, 27].

6.4. Rate functions of oriented crystallization

of polymorphic structures

The oriented crystallization coefficients of the ratefunctions derived in the above sections concerntransformation of the oriented amorphous phase intoa single crystalline structure. The coefficients de-rived for each of the transformation mechanism con-trolled by predetermined or sporadic nucleation areexpressed in terms of the parameters specific to thesingle polymorph, such as melting heat ∆hm, equi-librium melting point under zero stresses Tm

0, surfacefree energy densities σ and σe of the crystalline formsand thickness of the surface nucleus of crystalgrowth b. The parameters differ for different crys-talline polymorphs possible to grow in the systemsubjected to amorphous orientation.With the assumption that the other polymorphs donot interfere with the above parameters specified for


342

Figure 7. Reduced rate function K(T, fa)/K0(T, fa) vs. 103fa2for PET. Experimental points [11]: ∆T = 185 K (●),180 K (■), 175 K (), 170 K () and 165 K ();line plots: K(T, fa)/K0(T, fa) calculated from Equa-tion (29a) at ∆T = 185 K and 165 K.

each polymorph, we can consider transformationkinetics of each polymorphs in the system as inde-pendent of the other, except for the oriented amor-phous source phase being common for all of them.With the parameters specific for each polymorph, theinfluence of amorphous orientation on the crystal-lization rate function for each of them expresses bythe temperature-dependent oriented crystallizationcoefficients, Equations (28a), (29a). Important pointis to specify the temperature ranges at which orient-ed crystallization of the polymorph proceeds, i.e.below the equilibrium melting point for the poly-morph under the morphous orientation. With theranges of crystallization temperature assigned to eachof the polymorphs, the oriented crystallization ratefunctions of each polymorph can be calculated withthe oriented crystallization coefficients expressed byEquations (28a), (29a). The progress of crystalliza-tion in the oriented systems with a specified contri-bution of each polymorph can be calculated from acoupled set of kinetic equations of polymorphictransformation, present in the literature [72–74] andextended to oriented crystallization.The influence of amorphous orientation on the rangesof crystallization temperatures for particular poly-morphs is illustrated in Figures 8 where Gibbs freeenergy g is plotted schematically vs. temperature ina linear approximation near the equilibrium meltingpoints of the polymorphs in a three-phase systemcomposed of two crystalline polymorphs, ‘1’ and ‘2’,and oriented or unoriented (dashed line) amorphousphase. Entropy of the oriented amorphous phase is

decreased by the decrease in configurational entropyof the chains under the tensile stresses and the slopeof the free energy plot of oriented amorphous phasevs. temperature in Figures 8 is smaller than the slopeof the unoriented one.The tensile stresses do not influence entropy of thecrystalline polymorphs to such an extent as of theconfigurational entropy of the amorphous phase andthe slopes of the free energy plots of the polymorphsremain unchanged in this approximation. Entropies ofthe polymorphs differ and it is assumed that the poly-morph ‘1’ has the lower entropy in the example.Figure 8a illustrates the behavior of the systems inwhich the equilibrium melting point of the lower en-tropy polymorphic form ‘1’ is higher than that ofpolymorph ‘2’ in the unoriented state, Tm1

0 > Tm20 , and

Figure 8b – of the systems where Tm10 < Tm2

0 . The orderof the polymorphs melting points in the unorientedstate is associated with the ratio of the melting en-thalpy-to-entropy ratio, ∆hm

0/∆sm0, of the polymorphs.

In the case of systems characterized by the free en-ergy plots presented in Figure 8a, meta-stability ofthe higher enthalpy and/or higher entropy form ‘2’ ispreserved by the amorphous orientation in the entirerange of crystallization temperatures. The crystalliza-tion temperature range is shifted by amorphous ori-entation to higher values, and the apparent thermo-dynamic equilibrium between the polymorphs remainsabove the melting points of both crystalline forms.Such situation may be representative for melt spin-ning of polyolefins. In the case of isotactic polypropy-lene, thermodynamic conditions for crystallization


343

Figure 8. Schematic plots of the Gibbs free energy g vs. temperature in a three-phase system composed of two polymorphiccrystalline forms ‘1’, ‘2’ and oriented or unoriented (dashed line) amorphous phase. (a) stable polymorph ‘1’ andmeta-stable polymorph ‘2’ in the entire crystallization temperature range; (b) polymorph ‘1’ stable at low temper-atures and polymorph ‘2’ stable at high temperatures.

in the stable α and meta-stable β forms are preservedin the entire range of crystallization temperature inunoriented and oriented systems, like in Figure 8a.Figure 8b shows that at increasing melting heat ofthe high entropy polymorphic form ‘2’ (free energyplot is shifted down), thermal stabilities of the poly-morphs in the unoriented state are reversed at a hightemperature range. The polymorphic form ‘2’ is morestable than the polymorph ‘1’ in the high temperaturerange, above the temperature of thermodynamic equi-librium of the polymorphs, and becomes a singlecrystalline form above the equilibrium melting pointof the low entropy polymorph ‘1’. Amorphous orien-tation relatively does not influence the equilibriumpoint between the polymorphs, but it does extend therange of thermal stability of the higher entropy poly-morph and shifts the range to higher temperatures.This may concern also stabilization of mesophasesby stress-induced orientation in a high temperaturerange during high-speed melt spinning of polyestersor polyamides, followed by transformation to a morestable crystalline form at lower temperatures.

7. Conclusions

Crystallization half-time analysis of oriented crystal-lization [5, 27] indicates that the orientation effectsin the crystallization kinetics can be satisfactorily de-scribed by exponential function exp⎣A(T)fa2⎦ of thesingle quadratic term of the orientation factor fa mul-tiplied by the phenomenological coefficient A(T).Equation (3) adequately describes kinetics of orient-ed crystallization at low, as well as at high fa > 0.1values beyond the linear stress-orientation range. Ac-tually available models in the literature are not satis-factory for calculating the crystallization kinetics ofpolymers under variable orientation and temperaturewith the use of the exp⎣A(T)fa2⎦ formula due to thelack of a general expression for the function A(T).In the series expansion approach to the oriented crys-tallization kinetics we focus on the first expansionterm proportional to fa2 with the aim to derive formu-la for the temperature-dependent coefficient A(T)basing on the Hoffman-Lauritzen kinetic theory ofcrystal nucleation and growth. Our calculation arebased on the ‘proportional expansion’ approach [46]in the steady-state limit. The rate functions of orient-ed crystallization with separated sporadic and prede-termined nucleation mechanisms are calculated, aswell as their involvement in real systems subjected

to orienting stresses. The crystallization free energyaccounts for the entropy of uniaxial orientation of theamorphous chains. The orientation entropy is ex-panded in the power series over fa and the expansioncoefficients are calculated for Gaussian chains in thelinear stress-orientation range.Example computations of the oriented crystallizationcoefficient Ap(T) for crystallization with separatepredetermined nucleation and the ratio As(T)/Ap(T)of the coefficients for processes with separated spo-radic and predetermined nucleation are presented foriPP, PET, Nylon 66 and PLLA. High positive valuesof the coefficient are predicted in the entire range ofundercooling. The Ap values increase from severalhundred at large undercoolings to several thousandat decreasing the undercooling to 30 K. The Avramiexponent n has no influence on Ap. The predictedvalues of Ap for PET are of the order of the experi-mental values of the oriented crystallization coeffi-cient A determined for Equation (3) at several tem-peratures [11], and both values increase with increas-ing the temperature.The coefficient As(T) for processes with separatedsporadic nucleation is higher by a factor of about 4up to 7 at larger undercoolings and increases at ap-proaching the equilibrium melting temperature Tm

0.Higher values of As, in comparison to Ap, result fromhigher influence of orientation entropy on the freeenergy barrier of sporadic nucleation. Higher Avramiexponents only slightly reduce As. Relatively high As

values indicate that sporadic nucleation should beconsidered in the overall transformation kinetics athigher orientations. Example computations of thecrystallization rate functions for PET, Nylon 66 andPLLA with separated nucleation mechanisms indi-cate higher effects of amorphous orientation at lowerundercoolings.With the aim to estimate an involvement of sporadicand predetermined nucleation in real systems in theoverall crystallization kinetics, followed by crystalgrowth, we discuss the ratio of the oriented crystal-lization rate functions calculated as independent ki-netic characteristics. The conditions at which the in-volvement of sporadic nucleation approaches thelevel of the predetermined one in the kinetics are for-mulated. The involvement of sporadic nucleation inthe global kinetics, relative to the predeterminedone, is characterized by a product of an exponentialfunction responsible for the orientation effects and


344

a pre-exponential factor φ(T) which characterizesrelative contribution of the nucleation mechanismunder zero orientation, Equations (30) and (31).At low orientation, the main crystallization mecha-nism is associated with predetermined nucleationand the overall crystallization rate is controlled bythe predetermined nucleation, and A(T) ~= Ap(T). Therelative involvement of sporadic nucleation at smallorientations is reduced by many orders of the mag-nitude by thermodynamic factors associated with theundercooling.Reduction of the predetermined nuclei content N0 byone order of the magnitude results in an increase ofthe involvement of sporadic nucleation by one order.Marginal influence of the crystallite dimension R0 atwhich the crystal growth does not proceed is predict-ed. An increase of R0 between the critical value R*

and 10R* reduces the sporadic nucleation involve-ment by one order of the magnitude.Example computations of the crystallization ratefunction at small orientations for PET approximatedby the rate function controlled by predetermined nu-cleation are in qualitative agreement with the exper-imental results [11] and illustrate influence of the un-dercooling. We conclude that with increasing amor-phous orientation, involvement of both nucleationmechanisms may approach the same order at someorientation level and further increase of fa can leadto domination of sporadic nucleation. Basing on theseries expansion approach we formulate a hypothesisthat the increase of sporadic nucleation rate at in-creasing fa is responsible for an apparent agreementof the experimental observations [5, 27] of the crys-tallization kinetics at high amorphous orientationwith the single-term exp⎣A(T)fa2⎦ formula.Basing on the approach presented in this paper, theoriented crystallization rate functions can be formu-lated for all thermodynamically admissible transfor-mations between pairs of phases in polymorphic sys-tems subjected to tensile stresses, to be used in a setof kinetic equations of individual transformationsbetween the phases in the system.

AcknowledgementsThis research was supported by the basic funds of the Insti-tute of Fundamental Technological Research, Polish Acade-my of Sciences for scientific research.

References[1] Dumbleton J. H.: Spin orientation measurement in

polyethylene terephthalate. Textile Research Journal,40, 1035–1041 (1970).https://doi.org/10.1177/004051757004001111

[2] Ziabicki A.: Fundamentals of fiber formation.Wiley,London (1976).

[3] Ziabicki A., Kawai H.: High-speed fiber spinning.Wiley, New York (1985).

[4] Zieminski K. F., Spruiell J. E.: On-line studies and com-puter simulation of the melt spinning of nylon-66 fila-ments. Journal of Applied Polymer Science, 35, 2223–2245 (1988).https://doi.org/10.1002/app.1988.070350822

[5] Patel M., Spruiell J. E.: Crystallization kinetics duringpolymer processing – Analysis of available approachesfor process modeling. Polymer Engineering and Sci-ence, 31, 730–738 (1991).https://doi.org/10.1002/pen.760311008

[6] Ziabicki A., Jarecki L., Wasiak A.: Dynamic modellingof melt spinning. Computational and Theoretical Poly-mer Science, 8, 143–157 (1998).https://doi.org/10.1016/S1089-3156(98)00028-2

[7] Spruiell J. E.: Structure formation during melt spinning.in ‘Structure formation in polymeric fibers’ (ed.: SalemD. R.) Hanser, Munich, 5–93 (2000).

[8] Treloar L. R. G.: Crystallisation phenomena in raw rub-ber. Transactions of the Faraday Socciety, 37, 84–97(1941).https://doi.org/10.1039/TF9413700084

[9] Smith F. S., Steward R. D.: The crystallization of ori-ented poly(ethylene terephthalate). Polymer, 15, 283–286 (1974).https://doi.org/10.1016/0032-3861(74)90125-6

[10] Stein R. S.: Optical studies of the stress-induced crys-tallization of polymers. Polymer Engineering and Sci-ence, 16, 152–157 (1976).https://doi.org/10.1002/pen.760160306

[11] Alfonso G. C., Verdona M. P., Wasiak A.: Crystalliza-tion kinetics of oriented poly(ethylene terephthalate)from the glassy state. Polymer, 19, 711–716 (1978).https://doi.org/10.1016/0032-3861(78)90128-3

[12] Bragato G., Gianotti G.: High speed spinning of poly(ethylene terephthalate) – II: Orientation induced mech-anism of cold crystallization in pre-orientated yarns.European Polymer Journal, 19, 803–809 (1983).https://doi.org/10.1016/0014-3057(83)90150-7

[13] Desai P., Abhiraman A. S.: Crystallization in orientedpoly(ethylene terephthalate) fibers. I. Fundamental as-pects. Journal of Polymer Science Part B: PolymerPhisics, 23, 653–674 (1985).https://doi.org/10.1002/pol.1985.180230404

[14] Shimizu J., Okui N., Kikutani T.: Fine structure andphysical properties of fibers melt-spun at high speedsfrom various polymers. in ‘High-speed fiber spinning’(eds.: Ziabicki A., Kawai H.) Wiley, New York, 429–483 (1985).


345

https://doi.org/10.1016/S1089-3156(98)00028-2

https://doi.org/10.1016/0032-3861(74)90125-6

https://doi.org/10.1016/0032-3861(78)90128-3

https://doi.org/10.1016/0014-3057(83)90150-7

[15] Saijo K., Zhu Y-P., Hashimoto T., Wasiak A., Brzos-towski N.: Oriented crystallization of crosslinked cis-1,4-polybutadiene rubber. Journal of Applied PolymerScience, 105, 137–157 (2007).https://doi.org/10.1002/app.26019

[16] Flory P. J.: Thermodynamics of crystallization in highpolymers. I. Crystallization induced by stretching. Jour-nal of Chemical Physics, 15, 397–408 (1947).https://doi.org/10.1063/1.1746537

[17] Smith K. J., Greene A., Ciferri A: Crystallization understress and non-Gaussian behavior of macromolecularnetworks. Rubber Chemistry and Technology, 39, 685–711 (1966).https://doi.org/10.5254/1.3544874

[18] Krigbaum W. R., Roe R-J.: Diffraction study of crys-tallite orientation in a stretched polychloroprene vul-canizate. Journal of Polymer Science Part A: PolymerChemistry, 2, 4391–4414 (1964).https://doi.org/10.1002/pol.1964.100021010

[19] Kobayashi K., Nagasawa T.: Crystallization of shearedpolymer melts. Journal of Macromolecular Science,Part B: Physics, 4, 331–345 (1970).https://doi.org/10.1080/00222347008212506

[20] Yamamoto M., White J. L.: Theory of deformation andstrain-induced crystallization of an elastomeric networkpolymer. Journal of Polymer Science Part B: PolymerPhysics, 9, 1399–1455 (1971).https://doi.org/10.1002/pol.1971.160090804

[21] Gaylord R. J.: Orientation of crystallites formed instretched polymeric networks. Journal of Polymer Sci-ence Part C: Polymer Letters, 13, 337–240 (1975).https://doi.org/10.1002/pol.1975.130130604

[22] Ziabicki A., Jarecki L.: Theoretical analysis of orientedand non-isothermal crystallization III. Kinetics of crys-tal orientation. Colloid and Polymer Science, 256, 332–342 (1978).https://doi.org/10.1007/BF01544326

[23] Jarecki L., Ziabicki A.: Thermodynamically controlledcrystal orientation in stressed polymers. in ‘Flow-in-duced crystallization in polymer systems’ (ed.: MillerR. L.) Gordon and Breach, New York, 319–330 (1979).

[24] Katayama K., Yoon M. G.: Polymer crystallization inmelt spinning: Mathematical simulation. in ‘High speedfiber spinning’ (ed.: Ziabicki A., Kawai H.) Wiley, NewYork, 207–223 (1985).

[25] Ziabicki A., Jarecki L.: The theory of molecular orien-tation and oriented crystallization in high-speed spin-ning. in ‘High Speed Fiber Spinning’ (ed.: Ziabicki A.,Kawai H.) Wiley, New York, 225–269 (1985).

[26] Heuvel H. M., Huisman R.: Effect of winding speed onthe physical structure of as-spun poly(ethylene tereph-thalate) fibers, including orientation-induced crystal-lization. Journal of Applied Polymer Science, 22, 2229–2243 (1978).https://doi.org/10.1002/app.1978.070220815

[27] Patel R. M., Bheda J. H., Spruiell J. E.: Dynamics andstructure development during high-speed melt spinningof nylon 6. II. Mathematical modeling. Journal of Ap-plied Polymer Science, 42, 1671–1682 (1991).https://doi.org/10.1002/app.1991.070420622

[28] Ziabicki A., Jarecki L.: Crystallization-controlled lim-itations of melt spinning. Journal of Applied PolymerScience, 105, 215–223 (2007).https://doi.org/10.1002/app.26121

[29] Krieger I. M., Dougherty T. J.: A mechanism for non-newtonian flow in suspensions of rigid spheres. Journalof Rheology, 3, 137–152 (1959).https://doi.org/10.1122/1.548848

[30] Spruiell J. E., White J.: Structure development duringthe melt spinning of fibers. Applied Polymer Symposia,27, 121–157 (1975).

[31] Kulkarni J. A., Beris A. N.: A model for the necking phe-nomenon in high-speed fiber spinning based on flow-in-duced crystallization. Journal of Rheology, 42, 971–994(1998).https://doi.org/10.1122/1.550913

[32] Lin Y. G., Mallin D. T., Chien J. C. W., Winter H. H.:Dynamic mechanical measurements of crystallization in-duced gelation in thermoplastic elastomeric poly(propy-lene). Macromolecules, 24, 850–854 (1991).https://doi.org/10.1021/ma00004a006

[33] Titomanlio G., Speranza V., Brucato V.: On the simula-tion of thermoplastic injection molding process. Inter-national Polymer Processing, 12, 45–53 (1997).https://doi.org/10.3139/217.970045

[34] Pantani R., Speranza V., Titomanlio G.: Relevance ofcrystallisation kinetics in the simulation of the injectionmolding process. International Polymer Processing, 16,61–71 (2001).https://doi.org/10.3139/217.1620

[35] Floudas G., Hilliou L., Lellinger D., Alig I.: Shear-in-duced crystallization of poly(ε-caprolactone). 2. Evo-lution of birefringence and dichroism. Macromolecules,33, 6466–6472 (2000).https://doi.org/10.1021/ma000243b

[36] Doufas A. K., McHugh A. J., Miller C., Immaneni A.:Simulation of melt spinning including flow-inducedcrystallization: Part II. Quantitative comparisons withindustrial spinline data. Journal of Non-Newtonian FluidMechanics, 92, 81–103 (2000).https://doi.org/10.1016/S0377-0257(00)00089-6

[37] Tanner R. I.: A suspension model for low shear ratepolymer solidification. Journal of Non-Newtonian FluidMechanics, 102, 397–408 (2002).https://doi.org/10.1016/S0377-0257(01)00189-6

[38] Ziabicki A., Jarecki L., Sorrentino A.: The role of flow-induced crystallisation in melt spinning. e-Polymers,no. 072 (2004).https://doi.org/10.1515/epoly.2004.4.1.823


346

https://doi.org/10.1016/S0377-0257(00)00089-6

https://doi.org/10.1016/S0377-0257(01)00189-6

[39] Lee J. S., Shin D. M., Yung H. W., Hyun J. C.: Transientsolutions of the dynamics in low-speed fiber spinningprocess accompanied by flow-induced crystallization.Journal of Non-Newtonian Fluid Mechanics, 130, 110–116 (2005).https://doi.org/10.1016/j.jnnfm.2005.08.004

[40] Pantani R., Speranza V., Titomanlio G.: Simultaneousmorphological and rheological measurements on poly -propylene: Effect of crystallinity on viscoelastic param-eters. Journal of Rheology, 59, 377–390 (2015).https://doi.org/10.1122/1.4906121

[41] Jarecki L., Ziabicki A.: Mathematical modeling ofpneumatic melt spinning of isotactic polypropylene.Part II. Dynamic model of melt blowing. Fibers andTextiles in Eastern Europe, 16, 17–24 (2008).

[42] Jarecki L., Ziabicki A., Lewandowski Z., Blim A.: Dy-namics of air drawing in the melt blowing of nonwo-vens from isotactic polypropylene by computer model-ing. Journal of Applied Polymer Science, 119, 53–65(2011).https://doi.org/10.1002/app.31973

[43] Jarecki L., Blonski S., Blim A., Zachara A.: Modeling ofpneumatic melt spinning processes. Journal of AppliedPolymer Science, 125, 4402–4415 (2012).https://doi.org/10.1002/app.36575

[44] Jarecki L., Blonski S., Zachara A.: Modeling of pneu-matic melt drawing of poly-L-lactide fibers in the Lavalnozzle. Industrial Engineering and Chemistry Research,54, 10796–10810 (2015).https://doi.org/10.1021/acs.iecr.5b02375

[45] Nakamura K., Watanabe T., Katayama K., Amano T.:Some aspects of nonisothermal crystallization of poly-mers. I. Relationship between crystallization tempera-ture, crystallinity, and cooling conditions. Journal ofApplied Polymer Science, 16, 1077–1091 (1972).https://doi.org/10.1002/app.1972.070160503

[46] Ziabicki A.: Theoretical analysis of oriented and non-isothermal crystallization. II. Extension of the Kol-mogoroff-Avrami-Evans theory onto processes withvariable rates and mechanisms. Colloid and PolymerScience, 252, 433–447 (1974).https://doi.org/10.1007/BF01554749

[47] Ziabicki A: Theoretical analysis of oriented and nonisothermal crystallization. I. Phenomenological consid-erations. Isothermal crystallization accompanied by si-multaneous orientation or disorientation. Colloid andPolymer Science, 252, 207–221 (1974).https://doi.org/10.1007/BF01638101

[48] Fisher J. C., Hollomon J. H., Turnbull D.: Nucleation.Journal of Applied Physics, 19, 775–784 (1948).https://doi.org/10.1063/1.1698202

[49] Ziabicki A.: Generalized theory of nucleation kinetics.II. Athermal nucleation involving spherical clusters.Journal of Chemical Physics, 48, 4374–4380 (1968).https://doi.org/10.1063/1.1668003

[50] Kashchiev D.: Nucleation. Butterworth-Heinemann,Oxford (2000).

[51] Khana Y. P., Reimschuessel A. C.: Memory effects inpolymers. I. Orientational memory in the molten state;Its relationship to polymer structure and influence onrecrystallization rate and morphology. Journal of Ap-plied Polymer Science, 35, 2259–2268 (1988).https://doi.org/10.1002/app.1988.070350824

[52] Fillon B., Wittman J. C., Lotz B., Thierry A.: Self-nu-cleation and recrystallization of isotactic polypropylene(α phase) investigated by differential scanning calorime-try. Journal of Polymer Science Part B: Polymer Physics,31, 1383–1393 (1993).https://doi.org/10.1002/polb.1993.090311013

[53] Alfonso G. C., Scardigli P.: Melt memory effects in poly-mer crystallization. Macromolecular Symposia, 118,323–328 (1997).https://doi.org/10.1002/masy.19971180143

[54] Supaphol P., Spruiell J. E.: Crystalline memory effectsin isothermal crystallization of syndiotactic polypropy-lene. Journal of Applied Polymer Science, 75, 337–346(2000).https://doi.org/10.1002/(SICI)1097-

4628(20000118)75:3<337::AID-APP1>3.0.CO;2-4

[55] Ziabicki A., Misztal-Faraj B., Jarecki L.: Kinetic modelof non-isothermal crystal nucleation with transient andathermal effects. Journal of Materials Science, 51, 8935–8952 (2016).https://doi.org/10.1007/s10853-016-0145-8

[56] Ziabicki A.: Orientation mechanisms in the develop-ment of high-performance fibers. in ‘Orientational phe-nomena in polymers’ (eds.: Myasnikova L., MarikhinV. A.) Steinkopff, Frankfurt, Vol 92, 1–7 (1993).https://doi.org/10.1007/BFb0115443

[57] Ziabicki A., Jarecki L., Schoene A.: Transient biaxialorientation of flexible polymer chains in a wide rangeof deformation conditions. Polymer, 45, 5735–5742(2004).https://doi.org/10.1016/j.polymer.2004.05.070

[58] Ziabicki A., Alfonso G. C.: Memory effects in isother-mal crystallization. I. Theory. Colloid and Polymer Sci-ence, 272, 1027–1042 (1994).https://doi.org/10.1007/BF00652372

[59] Alfonso G. C., Ziabicki A.: Memory effects in isother-mal crystallization II. Isotactic polypropylene. Colloidand Polymer Science, 273, 317–323 (1995).https://doi.org/10.1007/BF00652344

[60] Schoene A., Ziabicki A., Jarecki L.: Transient uniaxialorientation of flexible polymer chains in a wide rangeof elongation rates. Polymer, 46, 3927–3935 (2005).https://doi.org/10.1016/j.polymer.2005.02.109

[61] Jarecki L.: Kinetic theory of crystal nucleation undertransient molecular orientation. in ‘Progress in under-standing of polymer crystallization’ (eds.: Reiter G.,Strobl G. R.) Springer, Berlin, 65–86 (2007).https://doi.org/10.1007/3-540-47307-6_4

[62] George H. H.: Spinline crystallization of polyethyleneterephthalate. in ‘High speed fiber spinning’ (eds.: Zi-abicki A., Kawai H.) Wiley, New York, 271–294 (1985).


347

https://doi.org/10.1002%2F(SICI)1097-4628(20000118)75%3A3%3C337%3A%3AAID-APP1%3E3.0.CO%3B2-4

[63] Hoffman J. D., Davis G. T., Lauritzen J. I. Jr.: The rateof crystallization of linear polymers with chain folding.in ‘Treatise on solid state chemistry’ (ed.: Hannay N.B.) Springer, Boston, 497–614 (1976).https://doi.org/10.1007/978-1-4684-2664-9_7

[64] Ziabicki A., Jarecki L.: Non-linear theory of stress-op-tical relations in polymer fluids. in ‘Progress and trendsin rheology II’ (eds.: Giesekus H., Hibberd M. F.)Springer, Heidelberg, 83–86 (1988).https://doi.org/10.1007/978-3-642-49337-9_18

[65] Salem D. R.: Draw-induced structure development inflexible-chain polymers. in ‘Structure formation in poly-meric fibers’ (ed.: Salem D. R.) Hanser, Munich, 118–184 (2001).

[66] Jarecki L., Misztal-Faraj B.: Non-linear stress-orienta-tion behavior of flexible chain polymers under fastelongational flow. European Polymer Journal, 95, 368–381 (2017).https://doi.org/10.1016/j.eurpolymj.2017.08.028

[67] Pȩcherski R. B.: Finite deformation plasticity with straininduced anisotropy and shear banding. Journal of Ma-terials Processing Technology, 60, 35–44 (1996).https://doi.org/10.1016/0924-0136(96)02305-9

[68] Pieczyska E. A., Pecherski R. B., Gadaj S. P., NowackiW. K., Nowak Z., Matyjewski M.: Experimental andtheoretical investigations of glass-fibre reinforced com-posite subjected to uniaxial compression for wide spec-trum of strain rates. Archive of Mechanics, 58, 273–291(2006).

[69] Jarecki L., Ziabicki A.: Thermodynamically controlledcrystal orientation in stressed polymers: 1. Effects ofstrain energy of crystals embedded in an uncrosslinkedamorphous matrix and hydrodynamic potential. Poly-mer, 18, 1015–1021 (1997).https://doi.org/10.1016/0032-3861(77)90005-2

[70] Jarecki L.: Thermodynamics of deformation of an iso-lated polymer chain. Colloid and Polymer Science, 257,711–719 (1979).https://doi.org/10.1007/BF01474099

[71] Jarecki L., Ziabicki A.: Development of molecular ori-entation and stress in biaxially deformed polymers. I.Affine deformation in a solid state. Polymer, 43, 2549–2559 (2002).https://doi.org/10.1016/S0032-3861(02)00031-9

[72] Ziabicki A., Misztal-Faraj B.: Modeling of phase tran-sitions in three-phase polymorphic systems: Part I. Basicequations and example simulation. Journal of MaterialsResearch, 26, 1586–1595 (2011).https://doi.org/10.1557/jmr.2011.195

[73] Misztal-Faraj B., Ziabicki A.: Modeling of phase tran-sitions in three-phase polymorphic systems: Part II. Ef-fects of material characteristics on transition rates. Jour-nal of Materials Research, 26, 1596–1604 (2011).https://doi.org/10.1557/jmr.2011.196

[74] Coccorullo I., Pantani R., Titomanlio G.: Crystallizationkinetics and solidified structure in iPP under high cool-ing rates. Polymer, 44, 307–318 (2003).https://doi.org/10.1016/S0032-3861(02)00762-0

[75] Janeschitz-Kriegl H.: Crystallization modalities in poly-mer melt processing. Fundamental aspects of structureformation. Springer, Wien (2010).https://doi.org/10.1007/978-3-211-87627-5


348

https://doi.org/10.1016/0924-0136(96)02305-9

https://doi.org/10.1016/0032-3861(77)90005-2

https://doi.org/10.1016/S0032-3861(02)00031-9

https://doi.org/10.1016/S0032-3861(02)00762-0

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