10584
J. Phys. Chem. 1992, 96, 10584-10586
(14) Saenger, W. Principles o/Nuc/eic Acid Structure; Springer-Verlag: New York, 1984. (15) Resch, U.; Eychmuller, A.; Haase, M.; Weller, H. Longmuir 1992, 8, 2215.
(16)Chander,R.R.;Bisham,S.R.;Coffer,J.L.J.Chcm.Educ.,inpieac. (17) Youn, H.-C.; Baral, S.;Fender, J. H. J . Phys. Chem. 1988,92,6320. (18) Henglein, A.; Gutierrez, M. Ber. Bunsen-Ges. Phys. Chem. 1983,87,
852.
A Lattice Treatment of Crystalline Solvent-Amorphous Polymer Mixtures on Melting Point Depression Yoshio Hoei,*
-
S&S Japan Co., Ltd., 6- 1 18, Hamazoe-dori, Nagata- ku, Kobe, Hyogo 653, Japan
Kamo Yamaura, and Shuji MaCsuzawa Faculty of Textile Science and Technology, Shimhu University, 3- 15- 1 , Tokida, Ueda, Nagano 386, Japan (Received: July 13, 1992; I n Final Form: October 2, 1992)
The conventional melting (and freezing) point depression equation for solvent crystal-amorphous polymer binary mixtures is reasonably derived using a free energy of fusion which is obtained by modifying Flory’s lattice theory of fusion for semicrystalline polymer-solvent systems. In consideration of the features of small molecular crystallites, a calculation of total configurational entropy is carried out assuming the “extended chain molecular crystal” model. For the heat of mixing, van Laar’s formula is used.
Introduction The classical melting (and freezing) point depression relationships of a crystallinesolvent for crystalline solvent-amorphous polymer mixtures are (l/Tm
- l/TOm)(ho/R)
=
-[In (1 - 4)
+ (1 - 1 / 4 4 + x4*1 (1)
and
-
- 1/Tm)(ho/R)
= -[ln (1 - 4) + 4 + X@l (x 0 3 ) (1’) Here T, and To, are the melting temperatures of the mixture and the pure solvent, respectively, ho is the molar heat of fusion, x is the polymer-solvent interaction parameter, x is the ratio of the molar volumes of the polymer and solvent, and $t is the volume fraction of the polymer. These classical relationships has been verified experimentally. These equations have been used to determine x values for a variety of polymer solutions’.* and to examine the departure of swollen polymer gel freezing points from the colligative property) and the effects of mixtures of crystalline species on the phase diagrams and eutectics.e7 On the other hand, the theoretical background for eqs 1 and 1’ was derived from the Flory-Huggins expression of chemical potential of mixing and the condition of phase equilibrium: Le., the chemical potential change of solvent in the crystalline phase is equal to that in the liquid (solution) phase in the equilibrium state.***In this paper, we will attempt to derive a more general expression for eq 1 which reduces eqs 1 and 1’ under certain limiting conditions. The derivation will be performed in terms of the degree of crystallinity and crystallite size on an analogy of Flory’s lattice theory of fusion for semicrystalline polymer-solvent mixtures9 (1/Tm
Model Our model concept of a single crystallite obeys that of an extended chain molecular crystal where all the molecules are exactly the same size. It has been established that this kind of crystal, for example, one composed of limited molecular weight n-alkanes, is formed.loJ’ To treat the present system in a lattice wherein all the segments of solvent molecules and polymeric *Towhom correspondence should be addressed. 0022-3654/92/2096-10584S03.00/0
structural units are assigned, it is assumed that bundles (i-e,, crystallites) of linear crystalline solvent sequences are of uniform length (tomolecules per sequence) randomly dispersed in the solution (of noncrystalline solvent and polymer) and each molecule (zo segments per molecule) is regularly placed side by side and end to end within the bundle, as illustrated in Figure la. The structural difference between our model and Flory’s fringed micelle model9 (see Figure 1b) is easily seen by comparison.
Method The parameters used are defined as follows: z is the number of segments of a polymeric structural unit, assuming that the size of a site (cell) forming the lattice is equal to that of a segment of the structural unit and solvent molecule; y is the number of nearest neighbors: No is the number of uniform solvent molecules; N is the number of uniform polymer molecules; x is the number of structural units per polymer molecule; is the total number of crystallite sequences; and wo = mto/Nois the degree of crystallinity. First of all, we estimate a total configurational entropy S, on the basis of the model structure in Figure la, which consists of four different entropies as shown by Flory? If we suppose a single long liiear chain randomly connected with all of the polymer and solvent molecules, the total number of possible arrangements of all the molecules in the chain W,is W,= (No+ N)!/N,$N!, which yields the first entropy Next, the number of configurations for the chain must be -timated in the lattice which is comprised of (N,,zo + XZN)sites. In the amorphous region each segment can be succwiively placed, one by one, at any of (y - 1) neighboring sites; thus, it is possible to conveniently express the entropy contribution by employing the disorientation entropy per segment k In [(y - l)/e] as previously formulated in the Flory-Huggins polymer solution treatment.* In the crystalline region the first site of each sequence is occupied by a segment which is followed by a segment which still belongs to the amorphous region, and thus the fmt segment gains entropy. However, no choice of location can be taken for the next (t,,zo - 1) segments which occupy the second and subsequent sites until reaching the opposite end site. A segment adjacent to an oppaeite end one regains entropy upon entering the amorphous region. 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10585
Letters
+-
50
molecules or segments
5080
__I
probability is analogous to Flory’s? A second probability arises from considering that none of the crystalline molecular ends occupy any site adjacent to the end of a noncrystalline molecule which approximates to (No + N - moSo)/(No+ N). This term creates the second term of the right side of eq 1 . Hence, the overall probability W4is given by W4= [(No N - moto)/(No N)IN0” X
+
+
(a)
(b) Figure 1. Schematic representations of a single crystallite: (a) extended chain molecular crystal model of solvent and (b) fringed micelle model of polymer.
Hence, the configurational entropy of the whole lattice S2can be given by s2 = w o z o + XNZ - mo(toz0 - 111 In [(r- l)/eI (3) A third probability W3consists of three different probability terms. The first probability is assumed as
I [Nozo/ (Nozo + X W l ( 1 / Z o > P with Nozo >> mo, which implies that the first site (of the first molecule) of a sequence is occupied by a segment of any solvent molecule out of (N,yo + X N Z )possible segments and the first segment that occupies the first site is chosen from zo possible segments of the molecule containing the first segment. For the first site for the second molecule or for each of the subsequent molecules of the sequence to be occupied by a solvent segment gives (No - mo)zo/(N,,zo X N Z )possibilities; thus, the second probability for mo sequences becomes
... (No+ Nozo + XNZ - motozo NO + N - mot0 Nozo+ xNz - mo{Go M![ No+ N And the fourth entropy S4is expressed as s 4 =
(
Consequently, the total configurational entropy &(e) is the sum of SI, S2,and S, minus the dmnnection entropy S4;thus, Sc(m,J = SI+ S2+ S3- S,. Sc(mpO)must reduce to the Flory-Huggins expression for the polymer solution state.*v9 The entropy change of fusion AS,,, = Sc(mo=O)- Sc(mo)is necessary to give the following relationship Nozo + XNZ NO+ N oN+ -N mot0 ] + m o l n [ D o ( Nozo
- Wfozo
+ XNZ) ] +
+
[(No- mo)zo/(NGo+ x N z ) ] ~ ( f o - l ) The third and contributing term [No- mo(to- l ) ] / N oimplies that the respective sites for the (to- 1 ) sequence molecules are unoccupied by any solvent molecules other than the solvent molecules. Thus, the total probability for all of the remaining sequence molecules will be given by ( [ N o- moU0- l)l/No)w(fo-l) It should be noted that thii term is significantto be able to achieve a correct temperaturevolume fraction relation with respect to u,,and Summarizing these three terms,the overall probability W3that all the sequences contain no polymer units is given by
where so kzo In [(y - l ) / e ] and Do (y - l)/z&. so and Do represent the entropy of fusion per molecule and the nucleation parameter of a sequence, respectively. If eq 6 is multiplied by Avogadro’s number, then the entropy change of fusion can be rewritten as a function of wo and to ASm/No
woso
-W ~ O , ~ O )
(7)
where [1
+~
4
/
~
4
0
~
-~0040) ~ 0
/
~
~
1
w3 = Accordingly
The enthalpy of fusion AH,,, consists of the heat of the fusion NO No~o+ XNZ
Since the above -a priori” probabilities are created assuming the statistical independency of events, the present treatment is restricted to that for the lower degree of crystallinity. At this point all the ends of the individual molecules in the chain are still connected. Thus, they must be disconnected to a real system. The fourth entropy S4which results from disconnection is obtained by considering two different probabilities. One is the probability that two end occupying adjacent sites are connected. This
motoho(ho being heat of fusion per molecule), and the heat of mixing AHmi, is the result of the formation of new molecular interaction pairs among Werent species. Regarding the Werence between the volume fraction of the amorphous solvent under the premelting state 44 and that under the postmelting state 40,the heat of mixing AHmixbecomes Mmix BVo(z/zo)Nx(40 - $0’1 from van Laar’s formula where $4 = 40(l - uo)/(l - wodo) and Vois the molar volume of the solvent. Thus, the enthalpy of fusion per molecule AH, f No is AHm/No = woho + BVo[w0#/(1 - w040)l
(8)
~
~
10586 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 +o=o.z; wo'=o.a a
z
F
i
q
+,o.,
/
I
0
8; wo'=o.8
A
m
I
1
I
20
40
60
J 60
50'
Figure 2. Theoretical plot of (l/Tm- l/Tmo)(ho/R)against crystalline length {
