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On a general approach to calculating the degree of crystallinity
Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
On a general approach to calculating the degree of crystallinity J. Mo¨ller, K.I. Jacob* Georgia Institute of Technology, School of Textile and Fiber Engineering, Atlanta, GA 30332-0295, USA Received 19 February 1998; accepted 1 March 1999
Abstract Crystals nucleating at random positions are considered. A number of basic considerations lead to simple equations describing crystals that do not grow spherically but have different growth rates in different directions, as well as time dependent nucleation and growth rates. As applications, the respective equations are derived for two- and three-dimensional cases for thermal and athermal growth. A number of results obtained in the literature are derived as special cases, including Avrami’s original equation. It is shown that the degree of crystallinity at any time is highest for perfectly parallel convex crystals, for which case an analytical solution is given. The power laws for the variation of the degree of crystallinity with time remain the same. A discussion is given as to what quantity would have to be computed in order to take into account the effect of any mutual orientation of crystals. 䉷 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Degree of crystallinity
1. Introduction For the theoretical computation of degrees of crystallinity, Avrami’s equation is frequently used [1–9]. The volume fraction of a material transformed due to nucleation and growth process has been extensively analyzed under the name Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory [10,11]. Predictions using Avrami’s equation is generally in compliance with experiments, for example, in the case of polymer crystallization [12,13]. There are a number of other papers in the literature, cf. [14–25], which make use of Avrami’s formulation and equation to characterize crystallization of materials. The frequently referred derivation of Avrami’s equation is based on a work by Evans [4]. However, the equation derived there is based on the assumption that spherical growth fronts propagate with a constant velocity. In polymer crystallization experiments, crystal shapes other than spherulites are frequently observed, especially in stress induced crystallization where the growth rate is different in different directions. Hence, when determining the degree of crystallinity, the mutual orientations of these crystals need to be taken into * Corresponding author. Tel.: ⫹ 1-404-894-2490; fax: ⫹ 1-404894-8780. E-mail address: [email protected] (K.I. Jacob)
account [10]. Moreover, linear growth rates may not be constant, but could depend on the size of the existing crystals [11]. We show that the problem of determining the degree of crystallinity as carried out by Evans [4], can be split up into the formulation of an “independent wave model” and an analysis of the degree to which it is applicable to different cases. First, a derivation given by Evans is critically analyzed and simplified. We discuss the applicability of Evan’s derivation for non-spherical crystals and growth fronts with non-constant propagation speeds. A number of equations from the literature are reproduced by means of the equations given here, a possible way of computing the influence of mutual orientation of crystals will be shown.
2. Basic equations We will closely follow the description of Evans’ derivation of Avrami’s equation as given in [14]. New formulations will be incorporated where it is appropriate. The derivation of the expressions for the degree of crystallinity for a two-dimensional case begins with the consideration of a mathematically related problem, solved by Poisson in 1837. When raindrops fall randomly on the surface of water, one expanding circular wave is created per raindrop.
0022-3697/99/$ - see front matter 䉷 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00033-5
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
1714
The probability of a representative point P to have been passed by n waves at time t is described by the Poisson problem. The solution to this problem is given by the Poisson distribution
area. The respective equations for E for three-dimensional cases are Zvt t ⫺ r=vr2 dr pI ⴱ v3 t4 =3 9 E 4pI ⴱ
Pn t e⫺E En =n!;
and
1
where E is the expectation value for the number of waves for the time interval t. The water waves are identified with crystalline growth fronts, and the rain drops hitting the surface stand for nucleation events in crystallization. A statement on the degree of crystallinity is now obtained by realizing that P0 t e⫺E
2
is the probability of point P not being crossed by any wave, or growth front, indicating that the area around this point is still amorphous. As the area is composed of either amorphous or crystalline regions, the degree of crystallinity, v c, can be written as vc 1 ⫺ P0 t 1 ⫺ e⫺E :
3
Later, we will refer to this model as the independent wave model. Now, the expectation value E has to be determined. A continuous rain of constant intensity falling on a surface of water translates into a thermal nucleation in two dimensions. For a rate of droplets hitting per unit time and unit area, I ⴱ, the number of waves having reached a point from a distance of r at time t since the beginning of rain is dE 2p t ⫺ r=vI ⴱ r dr;
4
v is the speed of propagation of the waves. As t is counted from the beginning of the rain, every element at a distance r from P can contribute to the number of waves arriving in a period t ⫺ r/v. Integration over r yields Zvt E 2pI ⴱ t ⫺ r=vr dr pI ⴱ v2 t3 =3: 5 0
The respective expression for E for the case of athermal nucleation, corresponding to a gush of rain at t 0 with N drops falling per unit area, is Zvt E 2pN r dr pNv2 t2 : 6 0
0
E 4pN
Zvt 0
r2 dr 4pNv3 t3 =3:
10
3. Generalization, a first approach 3.1. Theory As a first step, the equations given by Evans are extended in order to include different linear growth rates in different directions from the center of the crystal. Considerations made by Evans are carefully analyzed leading to a different approach than those taken by Evans, and resulting in a simpler equation. The major part of the derivation and argumentation is still carried out for two dimensions. The generalization of the respective results for three-dimensional considerations is either given explicitly, or can be derived accordingly. For the validity of Eq. (4), only the statistical independence of events with respect to time and location of nucleation events is necessary; the assumption that the growth fronts be circular or spherical is not required. Eq. (4) has to take into account different orientations of the crystal with respect to the radius vector, r, leading from the initial nucleation center, P I, which is now the center of the reference crystal, to an arbitrary point P. If we stipulate one direction within a reference crystal which is in some way characteristic of every crystal, the angle between this direction and r shall be denoted by w . (If the crystal has a rotational symmetry, i, there are i different directions in which the properties are identical. In order to keep the expressions simple, it is assumed that in each crystal, one of these directions is chosen at random.) The only propagation speed that needs to be taken into consideration is v(w ), the speed of that part of the wave front travelling towards P. For example, in the case of a constant linear growth rate, the geometry of a crystal can always be described in the form
In two-dimensional considerations, the functional determinant in Eqs. (5) and (6) is given by
R t; w v w t ⫺ t0 ;
J r 2pr;
with the dependence v(w ) being the same for all crystals. In this case, the differential r dE r; dr; w; dw I ⴱ t ⫺ 12 dwJ r dr v w
7
which is the length of a perimeter at a distance r. This quantity, called Jacobian determinant, occurs when switching from the differential dx dy to the product of the differentials of radius and angle, dr and df , respectively, in an integral. In three-dimensional cases it has to be replaced by the surface content of a sphere at this distance, J r 4pr2 : ⴱ
8
I and N are quantities per unit volume instead of unit
11
gives the contribution to E from all crystals that have an orientation of their inherent direction to the radius vector between them and point P in the range w and w ⫹ dw , and are located in a shell of thickness dr at a radius r around the point P. Let P(r,w ) be the probability density of finding a crystal at a distance r from the point P at an orientation w .
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
As the probability of finding a crystal at a certain orientation w with respect to point P does not depend on r, P(r,w ) can be separated into P r; w P rP w:
13
From calculus of probability, P w lim lim Pⴱ w
14
R1 !0 R2 !∞
with
X
ⴱ
P w lim
i
Dw!0
Xi R1 ⬍ r ⱕ R2 ; w ⬍ w 0 ⱕ w ⫹ Dw X 15 Dw Xj R1 ⬍ r ⱕ R2 j
can be taken as a definition of P(w ) and provides a means for numerically computing this quantity. In Eq. (15), X is the event of encountering a crystal with the respective property in the range indicated in the brace. It means that all events of Xj(R1 ⬍ r ⱕ R2) have to be counted in the denominator, that part of them that is in the subset of these events characterized by Xi(R1 ⬍ r ⱕ R2, w ⬍ w 0 ⱕ w ⫹ Dw ) is counted in the numerator. The procedure can be carried out for arbitrary, but fixed (i.e. independent from w ) values for R1 and R2. It is important to mention this, because we are dealing with crystals that may have a preferential orientation with respect to one another. As the system is considered to be infinite, it is important to count crystals in spheres or shells thereof (and not any other shapes) since the orientation of this shape to the preferred orientation of the crystals would alter the result for P(w ). In the integrals that follow, integral limits for r equal to 0 and ∞ must be considered to be created by the limit transition indicated by Eq. (15). With this in mind we can write P r
Z2p 0
P r; w dw;
P w
Z∞ 0
P r; w dr:
16
By inserting Eq. (13) into Eq. (16), the normalization of probability 1
Z2p Z∞ 0
0
P r; w dr dw
Z2p Z∞ 0
0
P rP w dr dw
17
is turned into Z2p 0
P w dw 1;
Z∞ 0
P r dr 1;
18
i.e. the marginal probability densities are normalized separately. With the definition given by Eq. (14), Eq. (4) turns into I r dE I ⴱ 19 P w dwJ r dr t⫺ v w and Eq. (5) can be written as I Zv wt r E Iⴱ t⫺ J r drP w dw; v w 0
20
1715
or explicitly for the two-dimensional case E
pI ⴱ t3 Z2p P wv w2 dw: 3 0
21
The first general statement we obtain is that orientation does not influence the power in the time dependence of v c as given by Eq. (3). That holds for Eqs. (5)–(10), as we will see later on when evaluating the respective integrals. This result does not presently mean that v 0 represents a degree of crystallinity; we will have to address the applicability of the independent wave model to the respective cases. This problem will be dealt with later. We proceed with the determination of P(w ). It gives the probability of finding a crystal of a certain orientation with respect to the orientation of the vector leading from P to the location of the respective crystal P I. Point P is supposed to be representative, i.e. chosen randomly, which means that every possible point P and every possible point P I are to be taken into consideration. First select a large number of randomly chosen locations P and another large number of randomly chosen locations P I, and consider every pair (P,P I). We want to find P(w ), the mean value of P with respect to all positions of P. P has the same probability of being at any specific location in the system, which, for example, may be given by (x,x ⫹ dx; y, y ⫹ dy). In polar coordinates it means that the point P has an equal chance of being in any differential w ⫹ dw with a fixed value r, as seen from any crystal. This is true for every fixed radius r, hence, for any choice of integral limits in an integral over r, as well. Summation over all crystal locations P I does not alter this result, since it holds for each crystal separately. What we have just found shows that the probability of finding a pair (P,P I) where the crystal has a certain orientation with respect to the vector leading from P to P I, P(w ), is independent of w . Alternatively, if a point P is considered in the first place, then the respective sum over all crystals has to be taken. In order to make P, a representative point, the summation over all possible points P has to be taken for averaging. The integration is carried out over the same set of pairs (P,P I), each of which characterized by values w and r. Considering Eq. (14), the same events X, characterized by r and w , are counted and assigned to the respective sums in this equation, no matter which summation is carried out first, the one over all P or the one over all P I. This yields P w constant:
22
Its value can be found from the normalization condition. In three-dimensional cases, w stands for the solid angle, and the upper limit in Eq. (18) changes from 2p to 4p. Hence, Eq. (18) leads to P2d w
1 ; 2p
P3d w
1 : 4p
23
This is the second result of this paper. A possible orientation has no influence on P0(t) as given by Eqs. (2) or (3). None of Eqs. (5)–(10) are influenced by such an orientation.
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
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That means that within the framework of the independent wave model the degree of crystallinity does not depend on any orientation. This result holds not only for linear growth but for any kind of growth as will be shown in the next section. Since v depends on w , the integrations over w in Eqs. (5)–(10) yield Z2p 2d Ethermal pI ⴱ v2 t3 =3 ! I ⴱ t3 v w2 dw=6; 24 0
2d Eathermal pNv2 t2 ! Nt2
Z2p 0
3d Ethermal pI ⴱ v3 t4 =3 ! I ⴱ t4
v w2 dw=2;
Z4p
3d Eathermal 4pNv3 t3 =3 ! Nt3
0
v w3 dw=12;
Z4p 0
v w3 dw=3:
25 26
(6) as special cases. In principle it is possible to compute the respective expressions for three-dimensional crystals in this way, but we leave that to the end of the next section, where we will have a more effective equation at hand. Wienberg et al. [11] have considered a similar case for anisotropic crystal growth and arrived at several important conclusions. They found that the shielding effect (blocking crystalline growth and removal of potential nucleation sites by particles) reduces the crystalline rate and hence the Avrami Exponent (AE) of the original Avrami equation. This effect is more predominant for needle-like particles. Also, reduction in AE is larger for a prenucleated (athermal) case. We will address few similar issues briefly later in this paper.
27 4. Generalization, a second approach
3.2. An application
4.1. Theory
For a demonstration of the theoretical considerations presented above, we choose a two-dimensional case. As an example for the applicability of Eqs. (24)–(27), a model crystal in the form of an ellipse is considered. The direction within a crystal shall coincide with the major axis of the ellipse. As the growth velocity at any point on the crystal surface is in the same direction as the radius vector leading from the center of a crystal to this point of surface (and as its value is proportional to its length), the parameter representation of the growth velocity can be written as
The derivation that follows is an alternative approach, independent from the works of Avrami, Evans, and similar theoretical developments. It is more general, even than the first generalization. In the discussion, it is compared to the work of Avrami revealing some interesting new aspects. The problem of finding the degree of crystallinity can be approached by means of a Poisson distribution in a slightly different way than presented by Evans in Ref. [4]. Instead of considering propagating independent water waves and the number of waves having crossed a certain point, the coverage of a point by water waves having passed this point can be considered. We already know that the coverage is Poisson distributed with a certain expectation value. Instead of considering the propagating wave fronts, drops of ink falling on a metal plate (assuming ink of different drops to run on top of each other) might be more relevant. The number of layers of ink at some location would obviously coincide with the number of fronts having passed this point. It is easier to consider the number of layers of ink instead. Its average number is simply given by the sum of the area of all ink spots having fallen per unit area. Here, the area under consideration has to be large enough to allow proper averaging of the quantities for unit area. The same consideration, of course, holds in three dimensions. We obtain
v2y v2x ⫹ 1; a2 b2
28
where a and b are the linear growth rates in the eigendirections and with sin w vx =v w;
cos w vy =v w
29
as the components of v, which represents an ellipse in the velocity space. The directions x and y of the coordinate system belonging to the specified crystal are coupled to the orientation of the ellipse. Eq. (29) results in ab v w p : 2 2 b sin w ⫹ a2 cos w2
30
The integral in Eqs. (24) and (25) can be evaluated, resulting in Z2p 0
v2 w dw
Z2p 0
2 2
a b dw 2pab: b2 sin w2 ⫹ a2 cos w2 31
For the expectation values of thermal and athermal nucleation for this simple model crystal we find Ethermal pI ⴱ t3 ab=3;
Eathermal pNt2 ab:
32
For a b, i.e. spheroidal growth, we obtain Eqs. (5) and
X E t lim 2d
A0 !∞
A t ⫺ ti
i;0ⱕti ⬍t
A0
:
33
At (t ⫺ ti) represents the functional dependence of the area covered by a crystal at time t that was formed at time ti. Hence, the sum in Eq. (33) gives the sum of coverage of all crystals, the nucleation of which have taken place in the area A0. A0 is supposed to be much larger than the biggest individual crystal. The respective expression for three
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
Fig. 1. Two perfectly aligned convex crystals are shown. A growth front that hits another crystal is stopped. The part of the growth front of crystal 1 that was prevented from being created by the presence of crystal 2 lies completely in crystal 2, and vice versa. Hence, no shadowing occurs.
dimensions is E t lim 3d
X
V t ⫺ ti
i;0ⱕti ⬍t
V0 !∞
:
V0
34
The equations are effective since they are valid for all geometries and all time laws of growth. The differential formulation of these equations can be found in the following way. We generalize by not considering the history of the process of crystallization but just the present state at time t. All crystals can be sorted according to their sizes, and to each crystal can be assigned a natural number, i, which can be substituted in Eqs. (33) and (34). If the number of crystals under considerations is very large, the size increase of crystals over any small range of i, i.e. Di, is small. Hence, we can count all i‘s in Di, which is equal to Di itself, multiply each of the A with its respective Di, and sum over this product. This yields X A t; jDi j E2d t lim
A0 !∞
j
A0
:
35
The Di(j)’s have to be continuous in i. Otherwise, the partitioning in Eq. (35) is completely arbitrary. It may be
Fig. 2. Two crystals which are not aligned are shown. The part of the growth front of crystal 1 that was prevented from being formed by the presence of crystal 2 does not lie completely in the area occupied by crystal 2. In the example shown, the occupied area is overestimated by the tip to the left of crystal 2, which is in reality shadowed by it (see also Ref. [9]).
1717
bound to the size of the crystals or the time when they nucleated. After binding the Di(j)’s in such a way, the number of i in each Di(j) goes to infinity with the limit transition in Eq. (35). This allows the Di(j)’s to be chosen arbitrarily small, by which we arrive at the differential formulation. If all Di(j) are of the same size then we can drop the suffix j, and the Di‘s divided by A0 are turned into dN Z En t V n N dN; 36 which means the discrete summations in Eqs. (33) and (34) have become integral. V (n) are the crystal volumes in n dimensions per unit volume. N can be represented as a function of time. N(t 0 ) is the number of nuclei at any given time t 0 , and the number of nuclei formed during a certain time interval dt in the history of the process is (dN/dt)兩t 0 dt. Since all crystals are assumed to follow the same growth law, all crystals that formed at a certain time t 0 are of the same size at time t. That means that V (n) is a function of time at its nucleation, t 0 , and the current time, t, i.e. V (n)(t 0 ,t). If the growth conditions for all individual crystals are assumed to be independent of time explicitly, V (n) depends on the time difference only, V (n)(t ⫺ t 0 ). For this case, we find the general expression Zt dN tⴱ 0 ⴱ 0 dt ; E n t V n t ⫺ t 0 37 0 dtⴱ t t t ⫺ ∞ where dN/dt is the nucleation rate. Eq. (37) simply means that crystals which formed in the time interval t 0 and t 0 ⫹ Dt (dN(t)/dt兩t ⴱt 0 Dt numbers) have now grown to a size V (n)(t ⫺ t 0 ). The sum over all these crystal sizes, normalized by the size unit provided by the nucleation rate, gives the expectation value for the number of independently grown crystals at randomly chosen locations. Eq. (37) holds for thermal and athermal nucleation for any shapes and for any time laws of nucleation and growth. Finally, the orientation of crystals plays no role, it does not even occur in these equations. As was indicated in the last two sections, something remains to be said on the validity of an independent wave model on a specific problem, i.e. its applicability for finding the correct degree of crystallinity in a semi-crystalline material system. In the process of growth, impingement occurs when two crystals meet and stop each other from growing into their mutually occupied areas. Whether two growth fronts are considered stopped (as in real systems) or they are considered to overlap (as in the independent wave model) has no effect on the validity of the considerations—as long as the quantity P(0) in Eq. (1) is not influenced. If they are considered to have stopped on impingement, all P(n) for n ⬎ 1 from Eq. (1) (the model basis of which is the independent wave model) would have to be added to P(1) in the resulting distribution, which would describe the real “coverage” by crystals. All P(n) for n ⬎ 1 in this resulting distribution would be zero, and as in the real material no area of the same dimensionality as
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
1718
the crystals themselves can be a part of two or more crystals at the same time. P(0) would remain unaltered—as long as the waves considered in the independent wave model can continue to travel after mutual impingement only in areas already occupied by other crystals without leaving them. This condition is fulfilled if all crystals are convex and perfectly aligned in parallel and if the linear growth rate is independent of the crystal size. After one growth front impinges on another crystal, the growth front of this crystal that would have to be passed travels at a speed at least as large as the linear growth rate in this direction. That means that for these perfectly oriented crystals, the independent wave model is exact. If the crystals are, however, not parallel, and not spherical, one wave in the independent wave model could leave another crystal again by overtaking the slower growth front propagating in the same direction, as shown in Figs. 1 and 2. This effect, opposite to impingement is called shadowing or shielding, c.f. [9,10]. This wave could, with a certain probability, hit an area that was previously uncrystallized. As this cannot happen in the physical process, where the wave is stopped on impingement, the independent wave model overestimates the real value of v c(t). As v c(t) according to the independent wave model is independent from orientation, it can be generally stated that higher degrees of orientation and axial ratios of crystals closer to unity result in a higher degree of crystallinity. Eq. (37) supply the upper limit for vc and provide its exact value for spherical crystals and for perfectly oriented crystals by means of Eq. (3). As long as no shadowing occurs, Eq. (37) is exact. Eqs. (33) and (34) can be used to formulate respective equations explicitly for the case of thermal and athermal nucleation. In the case of athermal nucleation, all crystals start growing at t 0, with a nucleation rate of N. Therefore we obtain 2d Eathermal t NA t;
3d Eathermal t NV t:
38
For thermal nucleation, the sum in Eqs. (33) and (34) have turned into an integral, with the number of nucleations, i ⴱ, per unit area being equal at all time intervals dt 0 of equal duration. Thus, E2d tthermal I ⴱ E3d tthermal I ⴱ
Zt 0 t t 0 0
Zt 0 t t 0 0
A t ⫺ t 0 dt 0 ;
39
V t ⫺ t 0 dt 0 :
The application of these equations will be shown in the next section. 4.2. Applications of the second generalization In this section we use Eqs. (33)–(39) to derive some of the equations we already obtained for linear growth as test cases. After that, expressions are derived for three-dimensional ellipsoids, which we had postponed until now. As a
last application, an equation for nonlinear growth is also given. Let us first derive Eqs. (5)–(10) as special cases. In the case of linear growth in two-dimensions we find A t t2
Z2p v w2 dw; 2 0
40
with Eq. (38) one can show that 2d Eathermal t Nt2
Z2p v w2 dw: 2 0
41
For a v independent of w , 2d Eathermal t
pNt2 v2 ;
42
which is Eq. (6). For ellipses we immediately obtain 2d Eathermal t pNt2 ab:
43
For thermal nucleation and linear growth, Eq. (39) yields ⴱ E t2d thermal I
Zt 0 t t 0 0
I ⴱ t3 =3
t ⫺ t 0 2 dt 0
Z2p v w2 dw 2 0
Z2p v w2 dw: 2 0
44
For ellipses we get ⴱ E t2d thermal pI ab
t3 : 3
45
At this point, computation of the respective equations for the three-dimensional case will be reviewed. The general expressions are those given by Eqs. (26) and (27) in these two cases. In Eq. (34), the volume of a crystal V(t) enters as a parameter. For ellipsoids, the three principle axes can be written as at, bt and ct. For the volume of the ellipsoid we find V t 4pabct3 =3:
46
Applying this result to Eqs. (26) and (27) yield 3d Ethermal
1 ⴱ pI abct4 3
and
3d Eathermal
1 4pNabct3 : 3 47
Next, one special three-dimensional case of growth equation shall be treated, in which the radius of the crystal grows with the square root of time. For simplicity, we assume the crystal shapes to be spherical. We write s t r t c0 48 t0 as a general form of such a growth equation, c0 and t0 being constants. For the crystal volume we obtain V t
4 4pc30 t3=2 pr t3 : 3 3t03=2
49
Here t0 and c0 are constants. From Eq. (39) we immediately obtain the respective equation for athermal
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
nucleation, 3d Eathermal t N
4pc30 t3=2 3t03=2
:
50
With dRc(u )/du v(u ), we have !d !d⫺1 d Rc t ⫺ Rc t 0 Rc t ⫺ Rc t 0 ⫺d v t 0 : 59 dt 0 a a
Eq. (38) gives the respective expression for the case of thermal nucleation, 0 4I ⴱ pc30 Zt t 8t5=2 I ⴱ pc30 0 3=2 0 t ⫺ t dt : 51 3t03=2 15t03=2 t 0 0
E t3d thermal
Ozawa’s [8] work can be used to compare our results, where a non-isothermal growth with thermal nucleation and cooling rate a was considered. Hence the temperature is a function of time. We compare the equation for E given by Ozawa with our results. Integral limits can be replaced, depending on the integration quantity used. The linear growth between time t and t is Zt r n T t 0 dt 0 ; 52
1719
For Ozawa’s form of the expectation value, we can write E T t Zt 0 t 0 Tt Zt 00 t 0
gd
t 0 t 0 Tm
t 00 t 00 Tm
n t 00 dt 00
Rc t ⫺ Rc t 0 a
!d⫺1 v t 0 dt 0 :
60 Considering Eq. (59), we obtain E T t ⫺
0 0 00 0 gd Zt t Tt Zt t d n t 00 dt 00 ⴱ d t 0 t 0 Tm t 00 t 00 Tm dt
Rc t ⫺ Rc tⴱ a
!d
tⴱ t 0 dt 0 :
61
t
where v is the linear growth rate. Ozawa further defines the functions ZT Rc T v T 0 dT 0 53 Tm
This equation can be integrated by parts according to the rule Zb Zb uv 0 dt 0 uv兩ba ⫺ u 0 v dt 0 : 62 a
With
and N u
Zu Tm
0
0
n T dT :
54
Here, n is the nucleation rate per d-dimensional unit area. For the expectation value, Ozawa gives gd ZT E T m⫹2 N uRc T ⫺ Rc um v u du 55 a Tm gd and m are constants, depending on the dimensionality of the problem, d, g2 2p, g3 4p and m turns out to be always in d ⫺ 1. In his work, Ozawa assumes spherulitic growth. Taking all his abbreviations and using our formulation (Eq. (37)), E(T) could be written as E T
a
0 gd Zt T n t 0 Rc t ⫺ Rc t 0 d dt 0 : d t 0 t 0 Tm
56
The question is whether the two integrals given by Eqs. (55) and (56) are identical. We integrate Eq. (55) by parts. For that, we compute !d d Rc t ⫺ Rc t 0 dt 0 a R t ⫺ Rc t 0 d c a
!d⫺1
! dRc t 0 : ⫺ a dt 0
57
With dT adt, we get !d !d⫺1 d Rc t ⫺ Rc t 0 Rc t ⫺ Rc t 0 dRc u t 0 ⫺d : dt 0 a a du t 0 58
0
u U N t
Zt 00 t 0 t 00 t 00 Tm
00
00
n t dt ;
vU
Rc t ⫺ Rc t 0 a
!d ;
63 we immediately obtain " !d # R t T ⫺ Rc t 0 t uv兩ba : N t 0 c t 0 Tm : a
64
At the lower limit, u N(t(Tm)) 0, at the upper limit, v 0, hence, uv兩t0 0: R For ⫺ ba u 0 v dt 0 we get Eq. (56), cf. Eq. (64), ⫺
Zb a
u 0 v dt 0
0 gd Zt T n t 0 Rc t ⫺ Rc t 0 d dt 0 : d t 0 t 0 Tm
65
66
We conclude that both expressions for E(T) are identical. That means that the equation given by Ozawa can be reproduced as one case of application from our present formulation. 5. Discussion An alternative way of calculating the degree of crystallinity is discussed, which is an appealing alternative for a number of reasons. The quantity that enters in the equation is easy to interpret, it is the n-dimensional area all crystals together would occupy if they grew independent from each other, normalized to the system volume. Time can be
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introduced fairly easily, to consider thermal and athermal nucleation. The results obtained in the literature in this respect can all be reproduced. We compare our results with those of Evans [4]. We already mentioned that Evans makes a statistical consideration using a Poisson distribution, where an expectation value E was calculated. The degree of crystallinity in this case is given by Eq. (3). If we compare this equation with Eq. (28) in [2] along with Eq. (6) in [1], E in Evans work as well as in the present investigation is represented by V1ex in Refs. [1,2]. Eq. (6) of Ref. [1] is identical to Eq. (37) of our paper. Further, it is identical to V~ of [3], which can be also seen from Eq. (7) in [3], which is the same as vc in Evan’s work. This is corroborated by comparing Eq. (2) in [3] with Eq. (37) of our paper. It can be seen from the first approach and in the generalization presented in this paper, that V1ex and V~ are identical. A more recent publication related to this work is [8]. In another recent publication [13], the authors use an equation similar to Eq. (37), but without giving a proof for its validity. As the comparison to literature shows, the work we present gives a more transparent and general view of characterizing the degree of crystallization. The concept of shadowing allows an assessment of the limits of applicability of the independent wave model to different situations, and even allows a quantitative treatment of the error made by applying it, c.f. Fig. 2 [10,11]. As the general result, the cases for which the independent wave model gives the exact degree of crystallinity are identical to those in which no shadowing can occur. Further, in systems in which shadowing can occur, the independent wave model gives an upper bound for the degree of crystallinity, c.f. Figs. 1 and 2. As a consequence, vc as given by Eq. (3) gives correct values for the degree of crystallinity for those cases in which (a) all nonspherical crystals are convex, (b) have the same linear growth rate dependence on the direction within the respective crystals, and (c) are perfectly aligned. Spherulitic crystals represent an example. It follows that the poorer the alignment, the less is the degree of crystallinity. It also follows immediately for ellipsoidal crystals that lower excentricity corresponds to higher degree of crystallinity, a result to which the authors in [10,11] obtained independently. It can also be seen that the work by Weinberg et al. [10] provide the lower limits for the degree of crystallinity for ellipsoidal crystals with given eccentricities and constant growth rates, while the upper limit is provided by this work.
Acknowledgements The authors are grateful to the National Textile Center for financial support. References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
M. Avrami, J. Chem. Phys. 7 (1939) 1103. M. Avrami, J. Chem. Phys. 8 (1939) 212. M. Avrami, J. Chem. Phys. 9 (1940) 177. U.R. Evans, Trans. Faraday Soc. 41 (1945) 365. A.N. Kolmogorov, On the question of geometrical selection of crystals, Report of the Academy of Sciences of the USSR, 1949, LXV(5):681, in Russian. A.N. Kolmogorov, Izv. Akad. Nauk SSR, Ser. Fiz. 3 (1949) 355 in Russian. I.H. Hillier, J. Polym. Sci. A 3 (1940) 3067. T. Ozawa, Polymer 12 (1971) 150. M.P. Shepilov, D.S. Baik, J. Non-Cryst. Solids 171 (1994) 141. M.C. Weinberg, D.P. Birnie III, J. Non-Cryst. Solids 202 (1996) 290. M.C. Weinberg, D.P. Birnie III, V.A. Shneidman, J. NonCryst. Solids 219 (1997) 89. L. Mandelkern, Crystallization of Polymers, McGraw-Hill, New York, 1964. K. Nakamura, T. Watanabe, K. Katayama, T. Amanato, J. Appl. Polym. Sci. 16 (1972) 1077. B. Wunderlich, Macromolecular Physics, Volume 2: Crystal Nucleation, Growth, Annealing, Academic Press, 1976. K.F. Kelton, A.L. Greer, C.V. Thompson, J. Chem. Phys. 79 (1983) 6261. K.F. Kelton, A.L. Greer, J. Non-Cryst. Solids 79 (1986) 295. V.A. Shneidman, M.C. Weinberg, J. Non-Cryst. Solids 160 (1993) 89. V.A. Shneidman, M.C. Weinberg, J. Chem. Phys. 97 (1992) 3621. I. Gutzow, D. Kashchiev, I. Avramov, J. Non-Cryst. Solids 160 (1985) 89. J.W. Christian, The Theory of Transformations in Metals and Alloys, Pergamon Press, Oxford, 1981. H. Yinnon, D.R. Uhlmann, J. Non-Cryst. Solids 54 (1983) 253. R. Trivedi, V. Seetharaman, M.A. Eshelman, Metall. Trans. 22 (1991) 585. L.P. Tarabacv, A.Yu. Mashikhin, V.O. Esin, J. Cryst. Growth 114 (1991) 603. C. Ruessel, J. Non-Cryst. Solids 219 (1997) 212. H. Nada, Y. Furukawa, J. Cryst. Growth 169 (1996) 587.
On a general approach to calculating the degree of crystallinity J. Mo¨ller, K.I. Jacob* Georgia Institute of Technology, School of Textile and Fiber Engineering, Atlanta, GA 30332-0295, USA Received 19 February 1998; accepted 1 March 1999
Abstract Crystals nucleating at random positions are considered. A number of basic considerations lead to simple equations describing crystals that do not grow spherically but have different growth rates in different directions, as well as time dependent nucleation and growth rates. As applications, the respective equations are derived for two- and three-dimensional cases for thermal and athermal growth. A number of results obtained in the literature are derived as special cases, including Avrami’s original equation. It is shown that the degree of crystallinity at any time is highest for perfectly parallel convex crystals, for which case an analytical solution is given. The power laws for the variation of the degree of crystallinity with time remain the same. A discussion is given as to what quantity would have to be computed in order to take into account the effect of any mutual orientation of crystals. 䉷 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Degree of crystallinity
1. Introduction For the theoretical computation of degrees of crystallinity, Avrami’s equation is frequently used [1–9]. The volume fraction of a material transformed due to nucleation and growth process has been extensively analyzed under the name Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory [10,11]. Predictions using Avrami’s equation is generally in compliance with experiments, for example, in the case of polymer crystallization [12,13]. There are a number of other papers in the literature, cf. [14–25], which make use of Avrami’s formulation and equation to characterize crystallization of materials. The frequently referred derivation of Avrami’s equation is based on a work by Evans [4]. However, the equation derived there is based on the assumption that spherical growth fronts propagate with a constant velocity. In polymer crystallization experiments, crystal shapes other than spherulites are frequently observed, especially in stress induced crystallization where the growth rate is different in different directions. Hence, when determining the degree of crystallinity, the mutual orientations of these crystals need to be taken into * Corresponding author. Tel.: ⫹ 1-404-894-2490; fax: ⫹ 1-404894-8780. E-mail address: [email protected] (K.I. Jacob)
account [10]. Moreover, linear growth rates may not be constant, but could depend on the size of the existing crystals [11]. We show that the problem of determining the degree of crystallinity as carried out by Evans [4], can be split up into the formulation of an “independent wave model” and an analysis of the degree to which it is applicable to different cases. First, a derivation given by Evans is critically analyzed and simplified. We discuss the applicability of Evan’s derivation for non-spherical crystals and growth fronts with non-constant propagation speeds. A number of equations from the literature are reproduced by means of the equations given here, a possible way of computing the influence of mutual orientation of crystals will be shown.
2. Basic equations We will closely follow the description of Evans’ derivation of Avrami’s equation as given in [14]. New formulations will be incorporated where it is appropriate. The derivation of the expressions for the degree of crystallinity for a two-dimensional case begins with the consideration of a mathematically related problem, solved by Poisson in 1837. When raindrops fall randomly on the surface of water, one expanding circular wave is created per raindrop.
0022-3697/99/$ - see front matter 䉷 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00033-5
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
1714
The probability of a representative point P to have been passed by n waves at time t is described by the Poisson problem. The solution to this problem is given by the Poisson distribution
area. The respective equations for E for three-dimensional cases are Zvt t ⫺ r=vr2 dr pI ⴱ v3 t4 =3 9 E 4pI ⴱ
Pn t e⫺E En =n!;
and
1
where E is the expectation value for the number of waves for the time interval t. The water waves are identified with crystalline growth fronts, and the rain drops hitting the surface stand for nucleation events in crystallization. A statement on the degree of crystallinity is now obtained by realizing that P0 t e⫺E
2
is the probability of point P not being crossed by any wave, or growth front, indicating that the area around this point is still amorphous. As the area is composed of either amorphous or crystalline regions, the degree of crystallinity, v c, can be written as vc 1 ⫺ P0 t 1 ⫺ e⫺E :
3
Later, we will refer to this model as the independent wave model. Now, the expectation value E has to be determined. A continuous rain of constant intensity falling on a surface of water translates into a thermal nucleation in two dimensions. For a rate of droplets hitting per unit time and unit area, I ⴱ, the number of waves having reached a point from a distance of r at time t since the beginning of rain is dE 2p t ⫺ r=vI ⴱ r dr;
4
v is the speed of propagation of the waves. As t is counted from the beginning of the rain, every element at a distance r from P can contribute to the number of waves arriving in a period t ⫺ r/v. Integration over r yields Zvt E 2pI ⴱ t ⫺ r=vr dr pI ⴱ v2 t3 =3: 5 0
The respective expression for E for the case of athermal nucleation, corresponding to a gush of rain at t 0 with N drops falling per unit area, is Zvt E 2pN r dr pNv2 t2 : 6 0
0
E 4pN
Zvt 0
r2 dr 4pNv3 t3 =3:
10
3. Generalization, a first approach 3.1. Theory As a first step, the equations given by Evans are extended in order to include different linear growth rates in different directions from the center of the crystal. Considerations made by Evans are carefully analyzed leading to a different approach than those taken by Evans, and resulting in a simpler equation. The major part of the derivation and argumentation is still carried out for two dimensions. The generalization of the respective results for three-dimensional considerations is either given explicitly, or can be derived accordingly. For the validity of Eq. (4), only the statistical independence of events with respect to time and location of nucleation events is necessary; the assumption that the growth fronts be circular or spherical is not required. Eq. (4) has to take into account different orientations of the crystal with respect to the radius vector, r, leading from the initial nucleation center, P I, which is now the center of the reference crystal, to an arbitrary point P. If we stipulate one direction within a reference crystal which is in some way characteristic of every crystal, the angle between this direction and r shall be denoted by w . (If the crystal has a rotational symmetry, i, there are i different directions in which the properties are identical. In order to keep the expressions simple, it is assumed that in each crystal, one of these directions is chosen at random.) The only propagation speed that needs to be taken into consideration is v(w ), the speed of that part of the wave front travelling towards P. For example, in the case of a constant linear growth rate, the geometry of a crystal can always be described in the form
In two-dimensional considerations, the functional determinant in Eqs. (5) and (6) is given by
R t; w v w t ⫺ t0 ;
J r 2pr;
with the dependence v(w ) being the same for all crystals. In this case, the differential r dE r; dr; w; dw I ⴱ t ⫺ 12 dwJ r dr v w
7
which is the length of a perimeter at a distance r. This quantity, called Jacobian determinant, occurs when switching from the differential dx dy to the product of the differentials of radius and angle, dr and df , respectively, in an integral. In three-dimensional cases it has to be replaced by the surface content of a sphere at this distance, J r 4pr2 : ⴱ
8
I and N are quantities per unit volume instead of unit
11
gives the contribution to E from all crystals that have an orientation of their inherent direction to the radius vector between them and point P in the range w and w ⫹ dw , and are located in a shell of thickness dr at a radius r around the point P. Let P(r,w ) be the probability density of finding a crystal at a distance r from the point P at an orientation w .
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
As the probability of finding a crystal at a certain orientation w with respect to point P does not depend on r, P(r,w ) can be separated into P r; w P rP w:
13
From calculus of probability, P w lim lim Pⴱ w
14
R1 !0 R2 !∞
with
X
ⴱ
P w lim
i
Dw!0
Xi R1 ⬍ r ⱕ R2 ; w ⬍ w 0 ⱕ w ⫹ Dw X 15 Dw Xj R1 ⬍ r ⱕ R2 j
can be taken as a definition of P(w ) and provides a means for numerically computing this quantity. In Eq. (15), X is the event of encountering a crystal with the respective property in the range indicated in the brace. It means that all events of Xj(R1 ⬍ r ⱕ R2) have to be counted in the denominator, that part of them that is in the subset of these events characterized by Xi(R1 ⬍ r ⱕ R2, w ⬍ w 0 ⱕ w ⫹ Dw ) is counted in the numerator. The procedure can be carried out for arbitrary, but fixed (i.e. independent from w ) values for R1 and R2. It is important to mention this, because we are dealing with crystals that may have a preferential orientation with respect to one another. As the system is considered to be infinite, it is important to count crystals in spheres or shells thereof (and not any other shapes) since the orientation of this shape to the preferred orientation of the crystals would alter the result for P(w ). In the integrals that follow, integral limits for r equal to 0 and ∞ must be considered to be created by the limit transition indicated by Eq. (15). With this in mind we can write P r
Z2p 0
P r; w dw;
P w
Z∞ 0
P r; w dr:
16
By inserting Eq. (13) into Eq. (16), the normalization of probability 1
Z2p Z∞ 0
0
P r; w dr dw
Z2p Z∞ 0
0
P rP w dr dw
17
is turned into Z2p 0
P w dw 1;
Z∞ 0
P r dr 1;
18
i.e. the marginal probability densities are normalized separately. With the definition given by Eq. (14), Eq. (4) turns into I r dE I ⴱ 19 P w dwJ r dr t⫺ v w and Eq. (5) can be written as I Zv wt r E Iⴱ t⫺ J r drP w dw; v w 0
20
1715
or explicitly for the two-dimensional case E
pI ⴱ t3 Z2p P wv w2 dw: 3 0
21
The first general statement we obtain is that orientation does not influence the power in the time dependence of v c as given by Eq. (3). That holds for Eqs. (5)–(10), as we will see later on when evaluating the respective integrals. This result does not presently mean that v 0 represents a degree of crystallinity; we will have to address the applicability of the independent wave model to the respective cases. This problem will be dealt with later. We proceed with the determination of P(w ). It gives the probability of finding a crystal of a certain orientation with respect to the orientation of the vector leading from P to the location of the respective crystal P I. Point P is supposed to be representative, i.e. chosen randomly, which means that every possible point P and every possible point P I are to be taken into consideration. First select a large number of randomly chosen locations P and another large number of randomly chosen locations P I, and consider every pair (P,P I). We want to find P(w ), the mean value of P with respect to all positions of P. P has the same probability of being at any specific location in the system, which, for example, may be given by (x,x ⫹ dx; y, y ⫹ dy). In polar coordinates it means that the point P has an equal chance of being in any differential w ⫹ dw with a fixed value r, as seen from any crystal. This is true for every fixed radius r, hence, for any choice of integral limits in an integral over r, as well. Summation over all crystal locations P I does not alter this result, since it holds for each crystal separately. What we have just found shows that the probability of finding a pair (P,P I) where the crystal has a certain orientation with respect to the vector leading from P to P I, P(w ), is independent of w . Alternatively, if a point P is considered in the first place, then the respective sum over all crystals has to be taken. In order to make P, a representative point, the summation over all possible points P has to be taken for averaging. The integration is carried out over the same set of pairs (P,P I), each of which characterized by values w and r. Considering Eq. (14), the same events X, characterized by r and w , are counted and assigned to the respective sums in this equation, no matter which summation is carried out first, the one over all P or the one over all P I. This yields P w constant:
22
Its value can be found from the normalization condition. In three-dimensional cases, w stands for the solid angle, and the upper limit in Eq. (18) changes from 2p to 4p. Hence, Eq. (18) leads to P2d w
1 ; 2p
P3d w
1 : 4p
23
This is the second result of this paper. A possible orientation has no influence on P0(t) as given by Eqs. (2) or (3). None of Eqs. (5)–(10) are influenced by such an orientation.
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That means that within the framework of the independent wave model the degree of crystallinity does not depend on any orientation. This result holds not only for linear growth but for any kind of growth as will be shown in the next section. Since v depends on w , the integrations over w in Eqs. (5)–(10) yield Z2p 2d Ethermal pI ⴱ v2 t3 =3 ! I ⴱ t3 v w2 dw=6; 24 0
2d Eathermal pNv2 t2 ! Nt2
Z2p 0
3d Ethermal pI ⴱ v3 t4 =3 ! I ⴱ t4
v w2 dw=2;
Z4p
3d Eathermal 4pNv3 t3 =3 ! Nt3
0
v w3 dw=12;
Z4p 0
v w3 dw=3:
25 26
(6) as special cases. In principle it is possible to compute the respective expressions for three-dimensional crystals in this way, but we leave that to the end of the next section, where we will have a more effective equation at hand. Wienberg et al. [11] have considered a similar case for anisotropic crystal growth and arrived at several important conclusions. They found that the shielding effect (blocking crystalline growth and removal of potential nucleation sites by particles) reduces the crystalline rate and hence the Avrami Exponent (AE) of the original Avrami equation. This effect is more predominant for needle-like particles. Also, reduction in AE is larger for a prenucleated (athermal) case. We will address few similar issues briefly later in this paper.
27 4. Generalization, a second approach
3.2. An application
4.1. Theory
For a demonstration of the theoretical considerations presented above, we choose a two-dimensional case. As an example for the applicability of Eqs. (24)–(27), a model crystal in the form of an ellipse is considered. The direction within a crystal shall coincide with the major axis of the ellipse. As the growth velocity at any point on the crystal surface is in the same direction as the radius vector leading from the center of a crystal to this point of surface (and as its value is proportional to its length), the parameter representation of the growth velocity can be written as
The derivation that follows is an alternative approach, independent from the works of Avrami, Evans, and similar theoretical developments. It is more general, even than the first generalization. In the discussion, it is compared to the work of Avrami revealing some interesting new aspects. The problem of finding the degree of crystallinity can be approached by means of a Poisson distribution in a slightly different way than presented by Evans in Ref. [4]. Instead of considering propagating independent water waves and the number of waves having crossed a certain point, the coverage of a point by water waves having passed this point can be considered. We already know that the coverage is Poisson distributed with a certain expectation value. Instead of considering the propagating wave fronts, drops of ink falling on a metal plate (assuming ink of different drops to run on top of each other) might be more relevant. The number of layers of ink at some location would obviously coincide with the number of fronts having passed this point. It is easier to consider the number of layers of ink instead. Its average number is simply given by the sum of the area of all ink spots having fallen per unit area. Here, the area under consideration has to be large enough to allow proper averaging of the quantities for unit area. The same consideration, of course, holds in three dimensions. We obtain
v2y v2x ⫹ 1; a2 b2
28
where a and b are the linear growth rates in the eigendirections and with sin w vx =v w;
cos w vy =v w
29
as the components of v, which represents an ellipse in the velocity space. The directions x and y of the coordinate system belonging to the specified crystal are coupled to the orientation of the ellipse. Eq. (29) results in ab v w p : 2 2 b sin w ⫹ a2 cos w2
30
The integral in Eqs. (24) and (25) can be evaluated, resulting in Z2p 0
v2 w dw
Z2p 0
2 2
a b dw 2pab: b2 sin w2 ⫹ a2 cos w2 31
For the expectation values of thermal and athermal nucleation for this simple model crystal we find Ethermal pI ⴱ t3 ab=3;
Eathermal pNt2 ab:
32
For a b, i.e. spheroidal growth, we obtain Eqs. (5) and
X E t lim 2d
A0 !∞
A t ⫺ ti
i;0ⱕti ⬍t
A0
:
33
At (t ⫺ ti) represents the functional dependence of the area covered by a crystal at time t that was formed at time ti. Hence, the sum in Eq. (33) gives the sum of coverage of all crystals, the nucleation of which have taken place in the area A0. A0 is supposed to be much larger than the biggest individual crystal. The respective expression for three
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
Fig. 1. Two perfectly aligned convex crystals are shown. A growth front that hits another crystal is stopped. The part of the growth front of crystal 1 that was prevented from being created by the presence of crystal 2 lies completely in crystal 2, and vice versa. Hence, no shadowing occurs.
dimensions is E t lim 3d
X
V t ⫺ ti
i;0ⱕti ⬍t
V0 !∞
:
V0
34
The equations are effective since they are valid for all geometries and all time laws of growth. The differential formulation of these equations can be found in the following way. We generalize by not considering the history of the process of crystallization but just the present state at time t. All crystals can be sorted according to their sizes, and to each crystal can be assigned a natural number, i, which can be substituted in Eqs. (33) and (34). If the number of crystals under considerations is very large, the size increase of crystals over any small range of i, i.e. Di, is small. Hence, we can count all i‘s in Di, which is equal to Di itself, multiply each of the A with its respective Di, and sum over this product. This yields X A t; jDi j E2d t lim
A0 !∞
j
A0
:
35
The Di(j)’s have to be continuous in i. Otherwise, the partitioning in Eq. (35) is completely arbitrary. It may be
Fig. 2. Two crystals which are not aligned are shown. The part of the growth front of crystal 1 that was prevented from being formed by the presence of crystal 2 does not lie completely in the area occupied by crystal 2. In the example shown, the occupied area is overestimated by the tip to the left of crystal 2, which is in reality shadowed by it (see also Ref. [9]).
1717
bound to the size of the crystals or the time when they nucleated. After binding the Di(j)’s in such a way, the number of i in each Di(j) goes to infinity with the limit transition in Eq. (35). This allows the Di(j)’s to be chosen arbitrarily small, by which we arrive at the differential formulation. If all Di(j) are of the same size then we can drop the suffix j, and the Di‘s divided by A0 are turned into dN Z En t V n N dN; 36 which means the discrete summations in Eqs. (33) and (34) have become integral. V (n) are the crystal volumes in n dimensions per unit volume. N can be represented as a function of time. N(t 0 ) is the number of nuclei at any given time t 0 , and the number of nuclei formed during a certain time interval dt in the history of the process is (dN/dt)兩t 0 dt. Since all crystals are assumed to follow the same growth law, all crystals that formed at a certain time t 0 are of the same size at time t. That means that V (n) is a function of time at its nucleation, t 0 , and the current time, t, i.e. V (n)(t 0 ,t). If the growth conditions for all individual crystals are assumed to be independent of time explicitly, V (n) depends on the time difference only, V (n)(t ⫺ t 0 ). For this case, we find the general expression Zt dN tⴱ 0 ⴱ 0 dt ; E n t V n t ⫺ t 0 37 0 dtⴱ t t t ⫺ ∞ where dN/dt is the nucleation rate. Eq. (37) simply means that crystals which formed in the time interval t 0 and t 0 ⫹ Dt (dN(t)/dt兩t ⴱt 0 Dt numbers) have now grown to a size V (n)(t ⫺ t 0 ). The sum over all these crystal sizes, normalized by the size unit provided by the nucleation rate, gives the expectation value for the number of independently grown crystals at randomly chosen locations. Eq. (37) holds for thermal and athermal nucleation for any shapes and for any time laws of nucleation and growth. Finally, the orientation of crystals plays no role, it does not even occur in these equations. As was indicated in the last two sections, something remains to be said on the validity of an independent wave model on a specific problem, i.e. its applicability for finding the correct degree of crystallinity in a semi-crystalline material system. In the process of growth, impingement occurs when two crystals meet and stop each other from growing into their mutually occupied areas. Whether two growth fronts are considered stopped (as in real systems) or they are considered to overlap (as in the independent wave model) has no effect on the validity of the considerations—as long as the quantity P(0) in Eq. (1) is not influenced. If they are considered to have stopped on impingement, all P(n) for n ⬎ 1 from Eq. (1) (the model basis of which is the independent wave model) would have to be added to P(1) in the resulting distribution, which would describe the real “coverage” by crystals. All P(n) for n ⬎ 1 in this resulting distribution would be zero, and as in the real material no area of the same dimensionality as
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
1718
the crystals themselves can be a part of two or more crystals at the same time. P(0) would remain unaltered—as long as the waves considered in the independent wave model can continue to travel after mutual impingement only in areas already occupied by other crystals without leaving them. This condition is fulfilled if all crystals are convex and perfectly aligned in parallel and if the linear growth rate is independent of the crystal size. After one growth front impinges on another crystal, the growth front of this crystal that would have to be passed travels at a speed at least as large as the linear growth rate in this direction. That means that for these perfectly oriented crystals, the independent wave model is exact. If the crystals are, however, not parallel, and not spherical, one wave in the independent wave model could leave another crystal again by overtaking the slower growth front propagating in the same direction, as shown in Figs. 1 and 2. This effect, opposite to impingement is called shadowing or shielding, c.f. [9,10]. This wave could, with a certain probability, hit an area that was previously uncrystallized. As this cannot happen in the physical process, where the wave is stopped on impingement, the independent wave model overestimates the real value of v c(t). As v c(t) according to the independent wave model is independent from orientation, it can be generally stated that higher degrees of orientation and axial ratios of crystals closer to unity result in a higher degree of crystallinity. Eq. (37) supply the upper limit for vc and provide its exact value for spherical crystals and for perfectly oriented crystals by means of Eq. (3). As long as no shadowing occurs, Eq. (37) is exact. Eqs. (33) and (34) can be used to formulate respective equations explicitly for the case of thermal and athermal nucleation. In the case of athermal nucleation, all crystals start growing at t 0, with a nucleation rate of N. Therefore we obtain 2d Eathermal t NA t;
3d Eathermal t NV t:
38
For thermal nucleation, the sum in Eqs. (33) and (34) have turned into an integral, with the number of nucleations, i ⴱ, per unit area being equal at all time intervals dt 0 of equal duration. Thus, E2d tthermal I ⴱ E3d tthermal I ⴱ
Zt 0 t t 0 0
Zt 0 t t 0 0
A t ⫺ t 0 dt 0 ;
39
V t ⫺ t 0 dt 0 :
The application of these equations will be shown in the next section. 4.2. Applications of the second generalization In this section we use Eqs. (33)–(39) to derive some of the equations we already obtained for linear growth as test cases. After that, expressions are derived for three-dimensional ellipsoids, which we had postponed until now. As a
last application, an equation for nonlinear growth is also given. Let us first derive Eqs. (5)–(10) as special cases. In the case of linear growth in two-dimensions we find A t t2
Z2p v w2 dw; 2 0
40
with Eq. (38) one can show that 2d Eathermal t Nt2
Z2p v w2 dw: 2 0
41
For a v independent of w , 2d Eathermal t
pNt2 v2 ;
42
which is Eq. (6). For ellipses we immediately obtain 2d Eathermal t pNt2 ab:
43
For thermal nucleation and linear growth, Eq. (39) yields ⴱ E t2d thermal I
Zt 0 t t 0 0
I ⴱ t3 =3
t ⫺ t 0 2 dt 0
Z2p v w2 dw 2 0
Z2p v w2 dw: 2 0
44
For ellipses we get ⴱ E t2d thermal pI ab
t3 : 3
45
At this point, computation of the respective equations for the three-dimensional case will be reviewed. The general expressions are those given by Eqs. (26) and (27) in these two cases. In Eq. (34), the volume of a crystal V(t) enters as a parameter. For ellipsoids, the three principle axes can be written as at, bt and ct. For the volume of the ellipsoid we find V t 4pabct3 =3:
46
Applying this result to Eqs. (26) and (27) yield 3d Ethermal
1 ⴱ pI abct4 3
and
3d Eathermal
1 4pNabct3 : 3 47
Next, one special three-dimensional case of growth equation shall be treated, in which the radius of the crystal grows with the square root of time. For simplicity, we assume the crystal shapes to be spherical. We write s t r t c0 48 t0 as a general form of such a growth equation, c0 and t0 being constants. For the crystal volume we obtain V t
4 4pc30 t3=2 pr t3 : 3 3t03=2
49
Here t0 and c0 are constants. From Eq. (39) we immediately obtain the respective equation for athermal
J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
nucleation, 3d Eathermal t N
4pc30 t3=2 3t03=2
:
50
With dRc(u )/du v(u ), we have !d !d⫺1 d Rc t ⫺ Rc t 0 Rc t ⫺ Rc t 0 ⫺d v t 0 : 59 dt 0 a a
Eq. (38) gives the respective expression for the case of thermal nucleation, 0 4I ⴱ pc30 Zt t 8t5=2 I ⴱ pc30 0 3=2 0 t ⫺ t dt : 51 3t03=2 15t03=2 t 0 0
E t3d thermal
Ozawa’s [8] work can be used to compare our results, where a non-isothermal growth with thermal nucleation and cooling rate a was considered. Hence the temperature is a function of time. We compare the equation for E given by Ozawa with our results. Integral limits can be replaced, depending on the integration quantity used. The linear growth between time t and t is Zt r n T t 0 dt 0 ; 52
1719
For Ozawa’s form of the expectation value, we can write E T t Zt 0 t 0 Tt Zt 00 t 0
gd
t 0 t 0 Tm
t 00 t 00 Tm
n t 00 dt 00
Rc t ⫺ Rc t 0 a
!d⫺1 v t 0 dt 0 :
60 Considering Eq. (59), we obtain E T t ⫺
0 0 00 0 gd Zt t Tt Zt t d n t 00 dt 00 ⴱ d t 0 t 0 Tm t 00 t 00 Tm dt
Rc t ⫺ Rc tⴱ a
!d
tⴱ t 0 dt 0 :
61
t
where v is the linear growth rate. Ozawa further defines the functions ZT Rc T v T 0 dT 0 53 Tm
This equation can be integrated by parts according to the rule Zb Zb uv 0 dt 0 uv兩ba ⫺ u 0 v dt 0 : 62 a
With
and N u
Zu Tm
0
0
n T dT :
54
Here, n is the nucleation rate per d-dimensional unit area. For the expectation value, Ozawa gives gd ZT E T m⫹2 N uRc T ⫺ Rc um v u du 55 a Tm gd and m are constants, depending on the dimensionality of the problem, d, g2 2p, g3 4p and m turns out to be always in d ⫺ 1. In his work, Ozawa assumes spherulitic growth. Taking all his abbreviations and using our formulation (Eq. (37)), E(T) could be written as E T
a
0 gd Zt T n t 0 Rc t ⫺ Rc t 0 d dt 0 : d t 0 t 0 Tm
56
The question is whether the two integrals given by Eqs. (55) and (56) are identical. We integrate Eq. (55) by parts. For that, we compute !d d Rc t ⫺ Rc t 0 dt 0 a R t ⫺ Rc t 0 d c a
!d⫺1
! dRc t 0 : ⫺ a dt 0
57
With dT adt, we get !d !d⫺1 d Rc t ⫺ Rc t 0 Rc t ⫺ Rc t 0 dRc u t 0 ⫺d : dt 0 a a du t 0 58
0
u U N t
Zt 00 t 0 t 00 t 00 Tm
00
00
n t dt ;
vU
Rc t ⫺ Rc t 0 a
!d ;
63 we immediately obtain " !d # R t T ⫺ Rc t 0 t uv兩ba : N t 0 c t 0 Tm : a
64
At the lower limit, u N(t(Tm)) 0, at the upper limit, v 0, hence, uv兩t0 0: R For ⫺ ba u 0 v dt 0 we get Eq. (56), cf. Eq. (64), ⫺
Zb a
u 0 v dt 0
0 gd Zt T n t 0 Rc t ⫺ Rc t 0 d dt 0 : d t 0 t 0 Tm
65
66
We conclude that both expressions for E(T) are identical. That means that the equation given by Ozawa can be reproduced as one case of application from our present formulation. 5. Discussion An alternative way of calculating the degree of crystallinity is discussed, which is an appealing alternative for a number of reasons. The quantity that enters in the equation is easy to interpret, it is the n-dimensional area all crystals together would occupy if they grew independent from each other, normalized to the system volume. Time can be
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J. Mo¨ller, K.I. Jacob / Journal of Physics and Chemistry of Solids 60 (1999) 1713–1720
introduced fairly easily, to consider thermal and athermal nucleation. The results obtained in the literature in this respect can all be reproduced. We compare our results with those of Evans [4]. We already mentioned that Evans makes a statistical consideration using a Poisson distribution, where an expectation value E was calculated. The degree of crystallinity in this case is given by Eq. (3). If we compare this equation with Eq. (28) in [2] along with Eq. (6) in [1], E in Evans work as well as in the present investigation is represented by V1ex in Refs. [1,2]. Eq. (6) of Ref. [1] is identical to Eq. (37) of our paper. Further, it is identical to V~ of [3], which can be also seen from Eq. (7) in [3], which is the same as vc in Evan’s work. This is corroborated by comparing Eq. (2) in [3] with Eq. (37) of our paper. It can be seen from the first approach and in the generalization presented in this paper, that V1ex and V~ are identical. A more recent publication related to this work is [8]. In another recent publication [13], the authors use an equation similar to Eq. (37), but without giving a proof for its validity. As the comparison to literature shows, the work we present gives a more transparent and general view of characterizing the degree of crystallization. The concept of shadowing allows an assessment of the limits of applicability of the independent wave model to different situations, and even allows a quantitative treatment of the error made by applying it, c.f. Fig. 2 [10,11]. As the general result, the cases for which the independent wave model gives the exact degree of crystallinity are identical to those in which no shadowing can occur. Further, in systems in which shadowing can occur, the independent wave model gives an upper bound for the degree of crystallinity, c.f. Figs. 1 and 2. As a consequence, vc as given by Eq. (3) gives correct values for the degree of crystallinity for those cases in which (a) all nonspherical crystals are convex, (b) have the same linear growth rate dependence on the direction within the respective crystals, and (c) are perfectly aligned. Spherulitic crystals represent an example. It follows that the poorer the alignment, the less is the degree of crystallinity. It also follows immediately for ellipsoidal crystals that lower excentricity corresponds to higher degree of crystallinity, a result to which the authors in [10,11] obtained independently. It can also be seen that the work by Weinberg et al. [10] provide the lower limits for the degree of crystallinity for ellipsoidal crystals with given eccentricities and constant growth rates, while the upper limit is provided by this work.
Acknowledgements The authors are grateful to the National Textile Center for financial support. References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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