J. Phys. Chem. 1995,99, 14024-14031
14024
Nonisothermal Crystallization of Polymers. 3. The Mathematical Description of the Final Spherulitic Pattern Ewa Piorkowska Centre of Molecular and Macromolecular Studies, Polish Academy of Sciences, 90-363 Lodz, Poland Received: November 15, 1994: In Final Form: May 16, 1995@
On the basis of the approach presented in the two preceding papers,'s2 the mathematical description of the final spherulitic pattems formed during nonisothermal crystallization in infinite samples is presented. The distributions of distances from spherulite centers to spherulite inner points, boundaries, and multiple boundary points are evaluated. Also the function describing the spatial correlation between the positions of the spherulite centers is obtained. The expressions for the total amount of interspherulitic boundaries and number of multiple boundary points are derived. The spherulitic structures are analyzed in detail for model modes of crystallization: (1) crystallization with instantaneous nucleation and (2) isokinetic crystallization. The parameters of structure resulting from nonisothermal crystallization of isotactic polypropylene are predicted and compared with the results of computer simulation of the spherulitic crystallization process.
Introduction
Background of the Derivations
A spherulitic structure is an important factor influencing many properties: impact and tensile properties (e.g. refs 3 and 4), kinetics of thermal degradation and gas sorption, (e.g. ref 4) among others. Especially the interspherulitic boundaries and multiple boundary points which constitute weak elements of spherulitic structure affect the destructive processes of polymers. The size and shape of a spherulite are determined by its nucleation time, the positions of neighboring centers, and their nucleation times and also by the process of growth. Hence, the spherulitic structure is determined by the primary nucleation process and the spherulite growth rate. The spherulitic pattems resulting from the isothermal crystallization processes were investigated experimentally (e.g. refs 7-9). Attempts were made to apply the methods used in quantitative microscopy for characterizing the spherulitic structures (e.g. ref 10). Also image analysis was applied for characterizing the spherulitic pattern in the thin However, the most information on the features of spherulitic structures was obtained by means of computer simulation for model modes of crystallization (e.g. refs 13-19). A mathematical description of the spherulitic structure formed during isothermal crystallization based on the concept of the primary nucleation being random in space and in time was developed previously by Piorkowska and Galeski.20,2' The possibility of application of this mathematical approach to more complicated crystallization conditions including nonisothermal ones was also pointed out in those papers. In the preceding the application of that concept to the mathematical description of spherulitic structure formation for time dependent conditions was presented. In this paper the final patterns of the spherulitic structure resulting from the nonisothermal crystallization process in infinite samples are mathematically described. The change of the conditions of crystallization is expressed by the time dependencies of the spherulite growth rate and the primary nucleation rate. The distributions of distances from the spherulite centers to spherulite inner points and boundaries as well as the correlation functions for the positions of spherulite centers will be derived.
The considerations that follow are based on the concept of the nucleation being random in space and in time presented in detail for the crystallization at time dependent conditions in the preceding papers.'.2 It should be mentioned here that according to those considerationsthe description of the nucleation process as occurring randomly in space but with an unfluctuated rate in time leads to the same results. However, it is less convenient from the mathematical point of view. The changes in crystallization conditions with time are reflected in the time dependencies of the spherulite growth rate G(t) and the nucleation rate F(t). As was shown previously only those circles, in a twodimensional case (2D), or spheres, in a three-dimensional case (3D), will occlude the arbitrarily chosen point A until time tm which are nucleated within the conical zone of space and time defined by the equation
@
Abstract published in Advance ACS Abstracts, July 1, 1995.
0022-365419512099-14024$09.0OlO
where r is the distance in space from point A and z denotes the time of the nucleation event. The circles (or spheres) nucleated at the surface of this zone, Le. at a distance r(t,tm)from point A, will occlude point A precisely at time t,. The first circle (or sphere) reaching a certain point in space is a real spherulite (i.e. its nucleation attempt occurred in an uncrystallized fraction of space) and is still growing at this point (Le. not truncated by the interspherulitic boundary). In the considerations that follow, the statement that a certain event occurs at the time t means that it occurs within the time interval t to t dt. The same concerns distances; r denotes the distance within the interval r to r dr. Since the probability that a nucleation event happens at a point within a space-time continuum equals zero, the infinitesimal volume elements resulting from multiplication of d g (2D) or d P (3D) and dt have to be considered.
+
+
Distribution of Distances from Spherulite Centers to Spherulite Inner Points If the arbitrarily chosen point A is occupied at time tm by only one growing circle (or sphere) nucleated at time z 5 tm 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 38, I995 14025
Description of the Final Spherulitic Pattem and none of the others nucleated until time t,, it becomes the internal point of a spherulite. The probability of such an event was derived in ref 1:
interval a to b from the spherulite centers. If the integration range is 0 to m, the result equals 1.
Distributions of Distances from Spherulite Centers to Spherulite Boundaries where AV(z,t,) is the volume of the annulus (or spherical shell) of radius r(t,tm) and E(tm)is the integral of the nucleation rate over the conical zone in the space-time continuum defined by eq 1:
In a similar way one can derive the distribution of distances of boundary points of n spherulites to spherulite centers. It was stated earlier',2 that, depending on the number of spherulites, n,and the dimensionalityof the sample, the spherulite boundary points constitute surfaces (3D n = 2), lines (3D n = 3 and 2D n = 2), and points (3D n = 4 and 2D n = 3). According to ref 2 the probability that an arbitrarily chosen point is occluded simultaneously at time tm by n spherulites nucleated at TI,t 2 , ..., z, and none of the others is expressed by the formula
where
and
for 2D and 3D cases, respectively. The numbers in parentheses denote the dimensionality. In order to calculate the probability of finding an inner point of a spherulite at the distance r from a center (equivalent to the demand that the arbitrary point A will be occupied first by the spherulite nucleated at the distance r from this point), one has to calculate the probability that the point A was occluded at time tm by a growing circle (or sphere) nucleated at the distance r = r(t,t,) and at time z and that no other circle (or sphere) reached this point until time tm. It follows from eq 2 that the probability of this event can be expressed as follows: pl*(2)(r,z)= 2nF(t)r e~p[-&~'(t,(r,z))] dz d r
(5a)
where AV(t,,t,) are defined by eqs 3a and 3b. For n = 1 eq 8a is equivalent to eq 2 and describes the formation of the spherulite interiors while for n > 1 it expresses the probability of the formation of interspherulitic boundaries. If the spherulites are nucleated at the same time z, the probability that n nucleation attempts will occur within the annulus (or spherical shell) AV(z,t,) has to be considered instead of the product of n component probabilities. Hence, the respective formula will be in the following form:
Equation 8b can be obtained from eq 8a by assuming zi = t for i = 1, ..., n and dividing the result by n!-the number of permutations of n spherulites. The relationships between the time of the simultaneous occlusion of point A by n spherulites, t,, the times of nucleation of those spherulites, TI, ..., tn,and the distances from their centers to point A, r ~ ..., , r,,, are as follows:
~ , * ( ~ ) ( r ,=z )4nF(z)r2 e~p[-E'~)(t,(r,z))]dz dr (5b) where tm(r,z) denotes the time of occlusion of point A by a spherulite as a function of the distance r and the time of spherulite nucleation z defined by the relationship
Equations 5 in fact describe the distributions of distances to spherulite inner points from the centers of spherulites nucleated at time z. Hence, integration over the range 0 r permits us to evaluate the sample fractions converted into spherulites nucleated at various time intervals. After the integration of the expressions' (eqs 5 ) right sides over the range 0 < z < -, one obtains the distributions of distances from inner points of all spherulites to their spherulite centers:
-
hI(')(r) dr = 2nr{J=F(t)
e~p[-&~'(t,(r,z))] dz} d r (7a)
l ~ ~ ( ~ dr ' ( r= ) 4 n ? { K F ( r ) exp[-d3'(t,(r,z))]
dz} dr (7b)
Integration of the distance distributions (eqs 7a and 7b) over the range a < r < b gives a sample fraction being in the distance
If the distance from point A to the center of a certain spherulite, which was nucleated at time t,equals r, the relationship (eq 9a) yields
where Tk and rk denote the nucleation times of the remaining n - 1 spherulites and the distances from their centers to point A , respectively. Relationships 9a and 9b permit us to express the time tm as a function of r and t,t,(r,r), and the distances rk as functions of Tk, r, and z, rk(q,r,z). Therefore, the probability that the boundary point of n spherulites, nucleated at times TI, ..., zn-l and z, is at the distance r from the center of a spherulite nucleated at t is expressed as follows:
14026 J. Phys. Chem., Vol. 99, No. 38, 1995
p,*(3)(r,z,tI ,...,T,,-J
Piorkow ska
=
h F ) ( r )dr = ( 4 ~ r ) ~ r ~ & ~e~p[-E'~)(t,(r,z))] F(t) x n- I
(4n)"F(T)r2 eXp[-E'2'(t,(r,T))]
[
dz dr" n F ( t k ) r:(tk,r,t) d t j
t(r'r) F(t,)r-(tj,r,z)
dtj] dz d? (14b)
k= 1
( 1Ob)
If the spherulites are nucleated at the same time z, the distances from their centers to their common boundary point are equal. Since that distance has to be counted n times, the respective probability is expressed by the equation
pns*(r,z)= p,*/(n - l)!
(1 1)
The factors d F ' in eqs 10 and in all distance distributions derived further represent the volume, area, or distance elements including boundaries between n spherulites. For n > 4 in the 3D case and n > 3 in the 2D case the probability of finding the boundary is negligibly small due to an excess of dr. For n = 2 eqs 10 yield
p2*'2'(r,z,zj) = 4 d F ( t ) F(z,)rr,(zj,r,z)e~p[-E'~)(f,(r,t))] dz, dz d? (12a)
Integrating h2(r) dr over the range 0 < r < and dividing by dr, one obtains the sum of all the spherulite boundaries in a unit area (or volume) of a sample; since each boundary point was counted twice (once per each spherulite), the obtained result has to be divided by a factor of 2. In a similar way, the formulas for the distributions of distances from spherulite centers to boundaries between three and four spherulites can be derived. In order to eliminate the result of multiple counting of spherulites upon integration over the range 0 < zj < t,, the resulting expressions have to be divided by (n - l)!. Hence OQ
h3(')(r)dr = 44r&-F(z) e~p[-E'~'(t,(r,z))l x [
[
4dF(t)r e~p[-E'~'(t,(r,z))]
r(r")
F(zj) rj(zj,r,z)dtj d t dr2 (13a)
f2(3'(r,z)d r dz = 16dF(z)? e~p[-E'~'(t,(r,z))l x
fl(r'r) F(zj)r-(zj,r,z)d t j dz drz (13b) Integrating expressions 13 over the range 0 < r < m and dividing them by the thickness of the boundary layer, dr, one obtains the area (in the 3D case) or the length (in the 2D case) of the boundaries of all spherulites nucleated at z per unit area (or volume) of a sample. Integration of expressions 13 over the range 0 < t < leads to the distance distributions for all boundaries between two spherulites which are expressed as follows: OQ
[
fl@")F(zj)r,(zj,r,z)dzj] dz dr2 (14a)
t(r'r) F ( t j ) r-(tj,r,z) dz$ dz d 3 (15b) [
t(r'r) r;(tj,r,z) dzjI3 dz d 4 (16) F(t,)
where A = 6 - ' ( 4 ~ ) ~ . Integrating h,(r) dr over the range 0 < r < and dividing by n d F 1 , one determines the number of multiple boundary points, the length of boundary lines, and the area of boundary surfaces, respectively, per unit area (or volume) of a sample. OQ
General Expressions for Distance Distributions On the basis of the above presented considerations, one can formulate the general formulas for the distance distributions from spherulite centers to spherulite boundaries and interiors. The distributions of distances from the centers of spherulites nucleated at z to spherulite inner points (n = 1) or boundaries (n > 1) are expressed by the formulas fn(2)(r,t) dr d t
f2'2'(r,z) d r dz =
x
l ~ ~ ' ~d' r( = r ) A r 2 L F ( t )e~p[-E'~)(t,(r,z))] x
Wb) Integration of pz*(r,z,zj) over the range a < tj < b leads to the distribution of probabilities of finding, at the distance r from the center of a spherulite nucleated at time z, the boundary with the spherulites nucleated within the time interval a to b: convex boundaries (with earlier nucleated spherulites), concave boundaries (with later nucleated spherulites), and boundaries in the form of a straight line or a plane formed with spherulites nucleated at the same time. The integration of relationships obtained in that way over the range 0 < z < m leads to the distributions of distances from spherulite centers to convex, concave, and straight boundaries, independently of spherulite nucleation times. After the integration of pz*(r,z,zj)over the range 0 < z < tm(r,Z) the distributions of distances from centers of spherulites nucleated at t to their boundaries with any other spherulites are obtained:
dzj]' d t d? (15a)
h3(3'(r)dr = 3 2 d ? L F ( z ) exp[-d3)(t,(r,t))]
p2*0)(r,z,zj)= 16dF(t)F(zj)[rrj(tj,r,z)] exp[ -E'3'( fm(r,t))] dzj d t dr2
t(r") F(zj)rj(zj,r,t)
= B,'2'[
r(r'r)
F(z,) rj(tj,r,z)dtjIn-I (17a)
where
B,'" = (n - 1)!-'(2n)"rF(z) e~p[-E'~'(t,(r,z>)] dz dr" and fn'3'(r,z) d r d t = BJ3'[ fl(r'r) F(zj) r:(zj,r,z) dzj]"-'
(17b)
where
= (n - 1)!-'(4n)"?F(t)
e~p[-E'~'(f,(r,z))] dz dl."
Expressions 17a and 17b can be written as a single formula: fn(r,z)d r dz = (n - l)!-'W(r) F(z) exp[-E(t,(r,z))]
where
x
J. Phys. Chem., Vol. 99, No. 38, 1995 14027
Description of the Final Spherulitic Pattem
Dividing the right sides of eqs 17 and 18 by dln-I, one obtains the amounts of the inner points (area or volume) and the boundaries (area, length, or number) at the distance r from the centers of spherulites nucleated at t per unit area (or volume) of a sample. Integration over the range 0 r < permits us to evaluate the area (or volume) or the total amount of boundaries of spherulites nucleated at t per unit area (or volume) of a sample:
Now, one can express the conditions in eqs 6 and 9 in the following forms:
R(t,) = r
R(tm)= r
+ y = rk+ zk
+y
(274
for k = 1, ...,n - 1 (27b)
where
Therefore eq 22 yields The formula describing the number of spherulites nucleated at time z per unit area (or volume) of a sample was derived in ref 1:
hi2)(r) d r = (n - l)!-'(2n)"rJWP(y) [L+'P(z)(r
N ( z ) dz = F(z) d t exp[-E(z)l
+ y)] x
+ y - z ) dz]"-'dy dr" (28a)
(21)
On the basis of formulas 20 and 2 1, one can easily evaluate the area (or the volume), the area and/or the length of boundaries, and the number of multiple boundary points for the average spherulite nucleated at a certain time t for 2D and 3D cases. Integration of the right sides of eqs 17 over the range 0 < t < 00 permits us to evaluate the distributions of distances from centers to the interiors and boundaries for all spherulites in a sample:
hi3)(r)d r = (n - 1)!-'(4n)'r2LmP(y) exp[-E,(3)(r [JhyP(z)(r
+ y - z ) dzI"-' ~
+ y)] x
dy dr" (28b)
Equations 28 can be expressed in a single form: h,(r) d r = W(r) dr"-'JmP(x-r)
h,(r) d r = Jmf,,(r,z) d t d r
[2 :r'
exp[-EN@)] -
dx dr (29)
(22) where P(z) = FN(z)/GN(z), F d R ) = F(t), GN(R) = G ( f ) ,W(r) is defined by eqs 19, and E N @ ) for the 2D and 3D cases, respectively, are expressed as follows:
Equation 22 yields h,(r) d r = (n - l)!-'W(r)JmF(t) exp[-E(t,(r,z))]
exp[--EN'2)(r
x
E
N
(2) R
(
- nJRP(z)(R - Z I 2 dz
(304
d z d r (23) Ek3'(R) = 4 / 3 ~ J R P ( ~ ) (R z ) dz ~ where W(r) is expressed by eqs 19a and 19b. The total area (or volume) of spherulites, the area and/or the length of boundaries, and the number of multiple boundary points per unit area (or volume) of a sample can be obtained in the following way:
(30b)
It follows from eqs 28-30 that the spherulitic pattem is fully determined by the ratio F(t)/G(t)expressed in terms of the extended spherulitic radius R ( f ) .
Distance Correlation
The average spherulite size, its boundaries' area and/or length, and the number of multiple boundary points can be evaluated as follows:
In refs 1 and 2 the amounts of the interspherulitic boundaries per unit area (or volume) of a sample were obtained by the integration of expression 8a over the ranges 0 < 21 < tmr..., 0 < z, < tm, and 0 < fm < -. However, integration over the ranges 0 < tl < tm, ..., 0 < zn-l < tm, 0 < t,,< -, and 0 < r < is equivalent and therefore leads to the same result. 00
It is often assumed that the centers of spherulites are distributed randomly over a sample because of the randomness of the nucleation process. However, the centers of spherulites in a sample crystallized isothermally are spatially correlated if nucleation is prolonged in time.*' In ref 21 the distance correlation function was defined as the ratio of the number of spherulite centers per unit area (or volume) at the distance r from an arbitrarily chosen center and scaled to the mean number of spherulite centers per unit area (or volume) in a sample. The aim of this section is to derive the formula describing the distance correlation between centers of spherulites nucleated during nonisothermal crystallization. The spherulite nucleated at z reaches the region at the distance r from its center at time fk defined by the equation
Extended Radius The distance distributions derived in the previous sections can be expressed in simpler forms by means of the introduction of the new variable called "extended defined by the equation
The fact that the said spherulite was nucleated at time t implies that no spherulites were nucleated at the distance r before the time to defined as follows:
14028 J. Phys. Chem., Vol. 99, No. 38, 1995
Piorkow ska
TABLE 1: Amounts of Intersphedtic Boundaries Surrounding an Average Spherulite for the Cases of Instantaneous Nucleation (IN) and Isokinetic Crystallization (ISP boundary form
Hence, for the nonisothermally crystallized sample the correlation function has the following form:
line point
Three-Dimensional Sample 2r(2/3)(4Jc/3)1~-~3 a 8 ~ ( 3 / 4 ) 4 / 3 ~ - 2 / 3 %4.37N-u3 %3.86N-Z3 4 r ( 1/3)(2n2/9)'/3N-"3 4 x 9 3 r(3/4)2]'/3N-'/3 x 13.92N-]I3 x 1 1.71N-I/) 12n 1on
surface
where N(t)dz is the number of spherulites nucleated at time z per unit area (or volume) of a sample, while tk and to are defined by eqs 31 and 32. The integrand in formula 32 represents the partial correlation function for the spherulites nucleated at time z. The substitution (eq 26) permits us to express C(r) in the form
IS
IN
Two-Dimensional Sample nN-l/2 121/4r(2/3)3/2~-1/* %3.14N-"* ~2.93N-'/~ 2n 2Jc
line point
N denotes the number of spherulites per unit area (or volume) of a sample.
where r(x) denotes the gamma function. The distance correlation function for the case of instantaneous nucleation is C(r) = 1
exp[-EN(x)l dx dz { J w P ( z ) exp[-EN(z)l
(34)
where z = R ( t ) and x = R(t). It follows that the correlation between spherulite centers is determined by the ratio of the nucleation rate to the growth rate expressed in terms of an extended radius.
The spherulitic pattern should not be affected by the change of crystallization condition only in the case of instantaneous nucleation of all spherulites. If the primary nucleation is prolonged in time the time dependencies of both nucleation rate and spherulite growth rate influence the resulting spherulitic structure. For the sake of demonstrating the influence of the nucleation mode on the final spherulitic pattern, two model modes of crystallization are considered: the instantaneous nucleation (all spherulites start to grow at the same moment) and the isokinetic crystallization (the ratio of the nucleation rate to the growth rate, P, is constant). The first case reflects the situation when self-seeded nuclei become active at a certain temperature. To some extent it may also describe the strong heterogeneous nucleation. The second considered case serves as an example of a nucleation process prolonged in time. In the case of the instantaneous nucleation the nucleation rate is expressed by the function F(t) = Dd(r), where d(r) denotes the Dirac delta function and D is the number of spherulite nuclei per unit volume (area) of a sample. The distributions of distances from the centers of spherulites to inner (n = 1) and boundary (n > 2) points are in the forms (followed from eqs 17 and 22)
h:3)(r)
For the isokinetic crystallization the distributions of distances from the centers of spherulites to their inner points (n = 1) and boundaries (n > 2) are in the following form (from eqs 17 and 22): h,,@)(r)d r = 2(n - l)!-1(~P)"r~wexp(-nPs3/3)s2"-2 ds d S (38a)
Model Modes of Nucleation
hL2)(r)d r = (n - 1)!-'(2nDr)" exp(-nDr2) dr"
(37)
hi3)(r)dr = 3(n - 1)!-'(4nP/3)"?~Bexp(-nPs4/3)s3n-3 ds d S (38b) The amounts of the spherulitic boundaries per unit area (or volume) of a sample are expressed in the following way (from eq 24):
PnT(2)= n!-1(9nP)("-')'3r[(2n
+ 1)/3]
(39a)
In the case of the isokinetic crystallization, the number of spherulites per unit area (or volume), N , equals
N'2)= ( ~ r ~ / 3 ) ~ / ~ n - ' r ( i / 3 )
N'3)=
(4w (40b)
Hence. formulas 39a and 39b assume the forms
(35a)
d r = ( n - 1)!-'(4nDr.')" exp(-4nDr3/3) dr" (35b)
The amounts of spherulite boundaries in a unit area (or volume) of a sample are expressed in the following way (followed from eq 24):
The area and/or length of interspherulitic boundaries and the number of multiple boundary points surrounding the average spherulite, Pa,, obtained according to formula 25 are listed in Table 1 for the crystallization with instantaneous nucleation and for the isokinetic crystallization. The distance correlation functions have the forms
J. Phys. Chem., Vol. 99, No. 38, 1995 14029
Description of the Final Spherulitic Pattern
1.0
1"
One can notice that
C ( r ) = O for r = O
(43a)
C(r) = 1 for r - m
(43b)
and Y
The distance distributions characterizing the spherulite pattems can be expressed in terms of y = rlRs, where R, denotes the radius of the average spherulite:
The distance distributions expressed in terms of y and normalized are in the forms Y
instantaneous nucleation
h,,@)(y)dy = 2r[(n h,,@)(y)dy = 3r[(2n
+ 1)/2]-'y"
exp(-y2) dy (45a)
+ 1)/3]-'y2" exp(-y3)
dy
(45b)
Figure 1. Normalized distributions of distances from spherulite centers to spherulite inner points (n = 1) and to boundaries between n spherulites (n = 2 and 3) as functions of a ratio y = r/Rs, where r denotes the distance and R, is the average spherulite radius, for twodimensional samples: a, instantaneous nucleation; b, isokinetic crystallization.
isokinetic crystallization
dy = i2r[(3n
+ 1)/41-'r(i/4)-(~"+')/~ Y X
A-exp[-z4r( 1/4)-4'3]~3"-3dz dy (46b) The distance correlation functions (eq 40) expressed in terms of y are in the forms 2.5
30
3.5
1
0
1.0
- 0.8 c
c r
0.6
0.4 0.2
As was expected from the relationships presented in this section, it follows that the spherulitic pattems are fully determined by the number of nuclei per unit area (or volume) of a sample, D, for the case of instantaneous nucleation. In the case of the isokinetic crystallization, the decisive parameter is the ratio of the nucleation rate to the spherulite growth rate, P. The distance distributions and the correlation functions in both cases considered do not depend on the nucleation rate and are identical to the respective dependencies presented for the isothermal crystallization in ref 21. If expressed as functions of the multiple of the radius of the average spherulite, y , the distance distributions and the correlation functions do not depend on D and P. Therefore they are typical for the nucleation mode and dimensionality of the sample.
0.0 0.0
0.5
1.0
1.5
2.0 2.5
3.0
3.5
Y Figure 2. Normalized distributions of distances from spherulite centers to spherulite inner points (n = 1) and to boundaries between n spherulites (n = 2-4) as functions of a ratio y = r/R,, where r denotes the distance and R, is the average spherulite radius, for threedimensional samples: a, instantaneous nucleation; b, isokinetic crystallization.
The distance distributions and the correlation functions expressed in terms of y are plotted in Figures 1-3 for the crystallization with instantaneous nucleation and for the isokinetic crystallization. From Figures 1 and 2 it follows that the
14030 J. Phys. Chem., Vol. 99, No. 38, 1995
Piorkowska spherulite growth rate
4
g(0 =
go e x p { - W V (..' ,* /
- VI-'} (48)
where go = 8009 cm s-l, U = 1500 cal mol-', Kg = 358 400 Kz, T, = 231.2 K, T," = 458.2 K, and
,,
:
: :'
T X ' } exp{-K$V',"
T(t) = To - a t
(49)
nucleation rate for T > 110 "C I
0.01. 00
I
.
0.5
.
4
I
1.0
1.5
2.0
2.5
'
I
F(t=O)At = C, exp(-C,To)
3.0
r Figure 3. Correlation functions C(y) between the positions of
spherulite centers for instantaneous nucleation in two-dimensional and three-dimensional samples (the straight solid line) and for isokinetic crystallization in two-dimensional (the dashed curve) and threedimensional (the dotted curve) samples vs a variable y = r/Rs,where r denotes the distance and R, is the average spherulite radius. maxima of the distributions of distances to boundaries are shifted with respect to the maxima of the distributions of distances to the spherulite inner points. The largest shift is in the case of distance distributions for triple (2D samples) and quadruple (3D samples) boundary points. The distance distributions for the isokinetic crystallization are broader and reach the maxima for greater values of argument than those for the respective distance distributions for instantaneously nucleated samples. This tendency is more pronounced in the case of the 3D crystallization. The data collected in Table 1 show that the instantaneous nucleation leads to the greater area and/or length of interspherulitic boundaries. The number of triple boundary points is the same in 2D samples for both model modes of crystallization considered, but the number of quadruple boundary points in 3D samples is greater for the case of instantaneous nucleation. The functions ?( y ) describing the spatial correlation between the positions of spherulite centers exhibit the same tendencies for both 2D and 3D cases. In the case of the instantaneous nucleation, the correlation function, E(y), equals 1; hence, the centers of spherulites are distributed randomly over a sample. In the case of the isokinetic crystallization, ?(y) starts from 0 and reaches the value of 1 for the values of y where the distributions of distances from spherulite centers to spherulite inner points and boundaries approach 0. Hence, in the close neighborhood of a certain spherulite center there is a relative lack of other spherulite centers. One can expect that the function P(y) will have a similar tendency for any time dependent nucleation process.
Crystallization of Isotactic Polypropylene As an example of a real crystallization process, the structural parameters in nonisothermally crystallized isotactic polypropylene (iPP) samples were computed. Similarly as in the preceding paper,2 the Rapra iPP (Mw = 307 O00, Mw/Mn= 20, density = 0.906 g cmw3,melt flow index = 3.9 g/10 min) was considered because the temperature dependencies of the spherulite growth rate and nucleation process are known for this polymer.22-28 Similarly as in the preceding paper? the crystallization processes during cooling, a, at rates 5, 10, and 20 Wmin were considered. The initial temperature, TO,equal to 132 "C for the beginning of calculations was chosen. The functions describing the time dependencies of the spherulite growth rate and nucleation rate were fitted as follow^:^,^^,^*
F(t)Ar = C,{exp[-C,T(t+At)]
- exp[-C,T(t)]}
(50) (51)
nucleation rate for 90 "C < T < 110 "C
F(t)At = C3[T(t+At)- T(t)]
(52)
where CI= 6215 x lo8 mm-3, CZ= 0.1567 K-I, and C3 = 3 167 K-' . The distance distributions from the spherulite inner points and boundaries to the centers of spherulites were numerically calculated using formulas 17b and 22. Also, the total area of the boundaries between pairs of spherulites, the total length of lines between the spherulite triads, and the number of boundary points between spherulite tetrads were computed for each rate of cooling considered. The normalized distance distributions expressed as functions of distance scaled to the average spherulite radius (36.1, 30.6, and 25.6 pm for cooling rates 5, 10, and 20 Wmin, respectively) are drawn in Figure 4. The points along the curve describing the distance distribution from centers of spherulites to their intemal points are the data from the computer simulation described in detail in ref 2. It is seen that the data for a certain type of distance distribution follow a single curve, independently of the cooling rate, as predicted theoretically by eqs 45 and 48. The good agreement between the theoretical predictions and computer simulation confirms the correctness of the mathematical approach developed in the paper. The obtained distance distributions are similar to those obtained for the isokinetic mode of crystallization (Figure 2b) and exhibit the same tendencies. The amounts of interspherulitic boundaries per 1 mm3 and those surrounding the average spherulite for all cooling rates considered are listed in Table 2. The amount of boundaries increases with an increase of the cooling rate, independently of the boundary type considered. The tendency reverses in the case of boundary surfaces and lines surrounding an average spherulite: the greater the cooling rate the smaller the surface surrounding an average spherulite. The total length of boundary lines between triads of spherulites also decreases. The number of boundary points between tetrads of spherulites on an average spherulite surface is almost independent of the cooling rate. The obtained data are similar to those listed in Table 1 for the isokinetic mode of crystallization. The reason for the similarities is that the variations of the ratio FIG with time for a given cooling rate are within a narrow limit of 30% after the onset of crystallization.
Conclusions The concept of the nucleation process as random events in space and time pennits us to describe the formation of the spherulitic structure during nonisothermal crystallization and also its final pattem. The final spherulitic structure can be characterized by the distributionsof distances from the spherulite centers to spherulite inner points and boundaries. There is a
J. Phys. Chem., Vol. 99, No. 38, 1995 14031
Description of the Final Spherulitic Pattern
1.0
n:l
0
1
2 3 4
1
1
3
r Figure 4. Normalized distribution of distances from spherulite centers to spherulite inner points (n = 1) and to boundaries between n spherulites (n = 2-4) as functions of the ratio y = r/R, for RAPRA iPP samples crystallized during cooling with the rates 5 , 10, and 20 Wmin. r and R, have the same meaning as in Figure 3.
TABLE 2: Amounts of Interspherulitic Boundaries in Nonisothermally Crystallized IPP Samples: (a) per 1 nun3of a Sample or (b) Surrounding an Average Spherulite" cooline rate 1Wmin) boundary form surface line point
a b a b a b
5
10
20
32.65 mm-l 0.0129 mm2 1112 mm-2 0.658" 37582 mm-3 29.65
38.50"-I 0.0093 mm2 1554 mm-2 0.562" 62496 mm-3 30.12
45.95 mm-' 0.0064 mm2 2229 mm-* 0.468" 108094 mm-3 30.25
The numbers of spherulites per 1 mm3 for cooling rates 5 , 10, and 20 Wmin are2 5070, 8300, and 14290, respectively.
possibility of calculating the total area andor length of interspherulitic boundaries and the number of multiple boundary points. Also the function characterizing the spatial correlation between the location of spherulite centers can be evaluated. Assuming a constant growth rate of spherulites, one obtains the relationships for the isothermal process. The spherulitic pattern is fully determined by the ratio of the nucleation rate to the spherulite growth rate F(r)lG(r) expressed in terms of the extended spherulite radius R(r). The consideration of the model modes of nucleation leads to the conclusion that in the case of instantaneous nucleation or isokinetic crystallization the resulting patterns do not depend on the variations of the spherulitic growth rate with time; important parameters are the number of spherulite centers per unit area (or volume) of a sample, D,for the instantaneous nucleation and the ratio of the nucleation rate to the spherulite growth rate, P , for the isokinetic crystallization. The analysis of the probability of formation of boundaries between spherulites nucleated at the same time leads to the conclusion that the pattern of broken straight line (or plane) boundaries is formed only if almost all spherulites are nucleated at the same time. Hence, such outlines of boundaries can serve as an indicator of an instantaneous nucleation process. If scaled to the radius of average spherulites, the spherulitic structure is characteristic for the mode of the crystallization process. It is possible to compare the results obtained for samples containing the same numbers of spherulites. The distance distributions are broader for the samples crystallized in the course of the isokinetic process, but the total length or
area of boundaries of an average spherulite is greater in the case of spherulitic structure nucleated instantaneously. Only in the case of instantaneously nucleated spherulites are their centers distributed randomly over a sample. The calculations conducted for nonisothermally crystallized samples of isotactic polypropylene cooled at various rates have shown that features of a final spherulitic pattern depend on cooling rate. The correctness of the analytical expressions was confirmed by the results of computer simulation. If the distance is scaled to an average spherulite radius, the respective distributions of distances become identical for various cooling rates and similar to those obtained for the isokinetic mode of crystallization due to a weak dependence of the ratio of the nucleation rate to the growth rate, F(r)lG(r),on time. Further development of the presented mathematical approach will allow for characterization of spherulitic structures formed under more complicated conditions-when the nucleation rate and spherulite growth rate depend also on space coordinates.
Acknowledgment. This research was supported primarily by the State Committee for Scientific Research through the Centre of Molecular and Macromolecular Studies, PAS, under Grant 2 P303 101 04. References and Notes Piorkowska, E. J. Phys. Chem. 1995, 99, 14007. Piorkowska, E. J. Phys. Chem. 1995, 99, 14016. Schultz, J. M. Polym. Eng. Sci. 1984, 24, 770. Mucha, M.; Kryszewski, M. Colloid Polym. Sci. 1980, 258, 743. (5) Galeski, A,; Piorkowska, E. J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 1299. (6) Galeski, A,; Piorkowska, E. J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 1313. (7) Misra, A.; Proudhomme, R. F.; Stein, R. S. J. Polym. Sci., Polym. Phys. Ed. 1974, 12, 1235. (8) Lovinger, A. J.; Chua, J. 0.; Gryte, C. C. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 641. 191 Pakula. T.: Galeski.. A.:. Piorkowska.. E.:. Krvszewski. M. Polvm. < Bull: 1979, 1 , 275. 110) Schulze. G. E.; Willers. R. J. Polvm. Sci., Parr B: Polvm. Phvs. 1987,25, 1311. (11) Tanaka, H.; Hayashi, T.; Nishi, T. J. Appl. Phys. 1986, 59, 653. (12) Tanaka, H.; Hayashi, T.; Nishi, T. J. Appl. Phys. 1986, 59, 3642. (13) Tabar, R. J.; Wasiak, A.; Hong, S. D.; Yuasa, T.; Stein, R. S. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 49. (14) Galeski, A. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 721. (15) Galeski, A.; Piorkowska, E. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 731. (16) Tabar, R. J.; Stein, R. S.; Rose, D. E. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 2059. (17) Billon, N.; Escleine, J. M.; Haudin, J. M. Colloid Polym. Sci. 1989, 267, 668. (18) Mehl, N. A.; Rebenfeld, L. J. Polym. Sci., Part B: Polym. Phys. 1993, 31, 1677. (19) Mehl, N. A.; Rebenfeld, L. J. Polym. Sci., Part B: Polym. Phys. 1993, 31, 1677. (20) Piorkowska, E.; Galeski, A. J. Phys. Chem. 1985, 89, 4700. (21) Piorkowska. E.; Galeski, A. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 1273. (22) Nakamura, K.; Katayama, K.; Amano, T. J. Appl. Polym. Sci. 1973, 17, 1031. (23) Martuscelli, E.; Silvestre, C.; Abate, G. Polymer 1982, 23, 229. (24) Clark, E. J.; Hoffman, J. D. Macromolecules 1984, 17, 878. (25) Bartczak, 2.;Galeski, A.; Pracella, M. Polymer 1986, 27, 537. (26) Galeski, A.; Bartczak, 2.;Pracella, M. Polymer 1984, 25, 1323. (27) Bartczak,Z . ; Galeski, A. Polymer 1990,31, 2027. (28) Galeski, A. In Polypropylene; Karger-Kocsis, J., Ed.; Chapman and Hall: London, 1994. (1) (2) (3) (4)
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