9498
J. Phys. Chem. 1992, 96, 9498-9507
Interchange Reactions during Polymerization: Reexamination of the Polymerization Mechanbm of ecaprolactam K. Suematsu* and T.Okamoto Kohno Medical Institute, Tomitahama 26-14, Yokkaichi City, Mie 512, Japan (Received: February 19, 1992)
The enumeration method is generalized to involve interchange phenomena. The solution is derived as a function of state: A chain, Nx,k,and a ring x-mer number, N R , ~in, state k are given in the forms k
+ 6NxQ)i-i
Nx, = Nx,a + 1-1 X
(10)
k
N R ~NR,O+ I-CI
(a&,
+ ~NR,Q)I-I
(26)
where the superscript Q interchange,so that the variations without Q represent variatio~based on polycondensation. The detailed structures of t h e variationsare shown in the text. With the help of the solution derived, a condition for simultaneous equilibrium between polycondensation and interchange is derived. The necessary and sufficient condition can be expressed by the chain rule kRJ/ kL = K',/K'
= KR/K
(44)
namely, the equivalency of the respective ratios of cyclization rate constants to corresponding interconnectionrate constants, where K'R, and xu are a cyclization and an interconnection rate constant, respectively, in U-B interchange, and KR and K are those in B-B interchange (U-B is short for unreacted functional unit-bond and B-B for bond-bond, respectively). bnder the condition of eq 44,all the reactions (the condensation and the interchanges) have a common equilibrium point whether ring formation occurs or not. With the aid of the present theory, the polymerization mechanism of ccaplactam is analyzed by taking all the rings into consideration. Making use of some fundamental relationships among rate constants, we have calculated respective values of the parameters; the results are compared with the clwical calculation. It is important to stress that some of the values found are still 'provisional". The reason is that there exists some basic difficulties to acquire exact values. This point will be discussed in detail in the text. A crucial point is if the more detailed experiments can be performed in a future study. We want to take a brief outlook on this problem. A theoretical prediction is offered for the B-B interchange.
Introduction This paper deals with interchange phenomena. More specifically, we want to extend the kinetic method introduced in preceding papers'**into a more general theory taking all possible interchange reactions into consideration in order to acquire deeper understanding of the dynamic behavior of linear polymer solutions. It is important to recognize that the interchange phenomena have been familiar phenomena: It has been known earlier among chemists that some types of interchange reactions occur under suitable conditions." Briefly, one type is the interchange between unreacted functional units (u.f.u.) and bonds (abbreviated as U-B interchange) while another is that between bonds (abbreviated as B-B interchange). There were positive proofs on polyamides: polyester^,^ polysiloxanes,6 polysulfides,' and Schiff-base" formation. Chromosome interchange in cells induced by X-ray irradiation9 and in cell divisions'" are well-known biological phenomena. The invasion of a virus into a human body and the release from the same are a typical B-B interchange between a cyclic molecule and a chain molecule.'" None of the changes of the number of molecules or bonds and the extent of reaction are observed. Instead, it has been supposed that the velocity toward an equilibrium state should be markedly accelerated. If this is the case,considering only bond formation through the condensation between u.f.u. must certainly fail in the exact description of real dynamics. Interchange phenomena have been discussed by some chemists, Flory" and Tobolsky,'* and recently by Lertola,I3 Kondepudi,14 Pojman,15 and co-workers. However, no one has so far offered the way of analyzing the behavior of cyclic spccic9 in irreversible processes. In what follows, a method is suggested which may cast a light on this problem. In sections I and 11, the theoretical consideration is taken, taking account of all the possible ring formations. In section 111, the 0022-3654/92/2096-9498$03.00/0
theory introduced is applied to the analysis of the polymerization mechanism of c-caprolactam.
I. Genenlizrrtion
Consider an AB model for simplicity in which a single molecule has two different functional units (fa.) of A and B,where A f.u. can react with B f.u., but not with A f.u. and the reverse is also true. The extension to an AA or an AA-BB model is straightforward, though slightly more complicated.12 As usual,we define the state in which k events occunrd as state k Chemical reactions associated with the interchange can be represented by the following five chemical equations: a. U-B interchange
c1
c2
Q 1992 American Chemical Society
x =I B
The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9499
Interchange Reactions during Polymerization b. B-B interchange
-+
Rx+j
3.
' c3
Rx
+ Rj
For j # x, there are (x j ) ways of forming one R,, while for j = x, there are x ways of forming two Ris. Hence, an increment of NR. becomes
~NR: C ( X + i)Qi-IIRx+j,Rx) j=1
4.
+ Rj
R,
-
(3)
Rx
The factor is required so that the same combinations are not counted twice. Hence
n
x- I
~ N R :E
C4
CQ,-i(Rj,L)(x -ANR,+i/hi-i
(4)
'I 1
R,'s death Lju
5.
~NR: c5 '
H,
R.
Lj
Assume the equal reactivity of f.u. and bonds, and a Markovian property for chemical processes. Let a state in which i events occurred be state i. We define the following transition probabilities, (Q], related to the interchange reactions:
U-B Interchange
Rj + R,
Q(XU,LI
+ i)u,Rxl
QWI 4. The probability that one bond in L,
interchanges with another bond in the same molecule to form R, Qlx+j,Rx)
QlRX,L) 6. The probability that one bond in R,, interchanges with another bond in the same ring, forming R, and Rj QlRx+j,Rx) where (Q]values with the superscript U express the U-B interchange, while those without U the B-B interchange. El. The Nmmbes of Ring x-mers, NRx. Assume an ideal chain. The variation of a ring x-mer number NR,in state ( i - 1 ) state i can be expressed as follows: R,)s birth
-
+
L,+ju R, Lj A partial increment of NRx is simply
~ N R =~ ZQi-ll(x+j)u,Rxl '
--
/=1
2.
L,+j
SNR: =
Co' j=1
(6)
R,+j
j- 1
NR,.i-i / h - i
(7)
If j = x - j , there are x/2 combinations, otherwise x combinations. Hence
6NR,"= - ( 1 / 2 ) ~xQi-l(R,,Rj)
(1)
(2)
for x 1 2
(8)
1- 1
Summing over all the partial variations, we have the total variation, 6NRQ,of the ring x-mer number. Namely, in the transition of (i - 1 ) i, we may write
-
~ N R ?Y
(9)
m
Combining eq 9 with another variation of eq A8 (Appendix 11) based on polycondensation,and then summing from i = 1 to k,we arrive at the final expression of the ring x-mer number in state k It is convenient to summarize the lengthy expression as k
NR,k
E
NR,O + Z [6NRI+ 6NRpl,-l i= 1
(10)
where
~ N R ~ JPi-ilx,RJ -~ - Pi-iLBI
(A81
1-2. The Number of Chain x-mem,N,,,.In a similar way, partial variations of a chain x-mer number N, can be derived as follows: L i s birth 1.
SN,' 2.
R, + Lj
1) Qi-1b+j),RXl
Eo' - 1)N,,i-l/hbl
Rx --* Rj + Rx-j
5. The probability that one bond in R, interchangeswith bonds in other molecules
-
LHj
j=l
+
(5)
x-1
B-B Interchange 3. The probability that one bond in L, interchanges with bonds in other molecules
1.
+
~NR; -Qi-ilRx,L)
2. The probability that an unreacted A or B f.u. on LHj reacts with one bond in the same molecule to form R, QKX
LHj
If j # x, one R, disappears, while ifj = x, two R,'s disappear, so that
8.
1 . The probability that an unreacted A or B f.u. on L, interchanges with bonds in other molecules
+ R,
-Qt-l(Rx,L]
7.
"xtj
J
-
-CQi-iCiU,L)xNR.,i-i/hi-i J-1
6.
6NR:
+ R,
LPjU Y
+ Lj+,
-
L,
+ L,
2 C Q,-l{(x-j)u,L) Nj+,,i-l/h,-l
j=lm=l
Lju + Lm+x
-
Lj+m + Lx
for x 1 2
(11)
Suematsu and Okamoto
9500 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992
Rx 6.
L,j
+ Rj
+
Z b - j - 1) Q,i-ll(x-j),L)jNRIJ-l/hi-l
I= 1
7.
Lx+j
-
for x L 3 (16) Lx
1.1
f
Rx-J
Figure 1. Schematic representation of internal double cyclization.
Lx
r l
6N*6 N-
+
1-3. Kioefk Exg" of T"&h ProbMWies (Q]. Within the infinitesimal time interval, the total number, dM,of the chemical events must be
dM+Adt
+ Rj
If the dangling end length is x, there are (x - 1) chances of
(27)
Let V be a system volume at a given state i - 1. The explicit expressions of the transition probabilities, may then be written in the kinetic forms
{a,
Qi-ilxU,L)Y [ ~ ~ N x , i - i b i / U / A Qi-il(x+j)',Rx) Qi-llx,LI
8.
[KUR,Nx+),i-ll/ A
(30)
[KXNR,i-lhi-I/VI /A
(31)
Qi-l(x+i,Rxl
9.
(29)
[fix,,i-lhi-~/VI / A
H
Qi-I(Rx,LJ
(28)
Qi-i(Rx+j&J
Y
[KR,Nx+j.j-d/ A
(32)
[PflR&j,i-lI/A
(33)
where 10. b-1
L-1
11.
L-2
is the rate constant of the internal double cyclization, R, Rj ,,R and has the following property
pj
12.
+
PI
3
~x-1= ( K R ~ J / K R ~
-
(35)
1-4. prob.Mity of Intcrnrl Cyclization. Equation 35 can be derived as follows: Let the distance between reaction points in an x-ring (12) be r. The probability that the ends of aj-chain in the ring are at the distance of r is aexp(-39/2(r 2)), while for an (x -j)-chain the probability is aexp(-39/2(rFj I )). Under the assumption of ideal chains, the probability that the ends of thejchaiu and the (x -j)-chain are simultanaously at the distana of r can be written as the product of them
13.
14.
Summing again over the partial variations, one obtains the total variation (see the proof in Appendix I)
Combination with the variation, 6Nx, of eq A9 based on polycondensation and subsequent rearrangement lead us to the expression of a chain x-mer number in state k. Hence we have obtained the distriitions of acyclic and cyclic species as a function of state k
NR,L
CI [ ~ N +R 6NRpli-1 ~ - NR,O + i=
(10)
k
N X L
= NX.0 + I= C1 [6Nx + 6NxQ1i-i
(26)
These equations can be solved numerically for any state with the help of hk and h given in Appendix 11.
wj,(r) = ex~(-39/2(r~,~)) = exp(-39 / 2 (r;) ) exp(-39 / 2 (r-;) ) On the other hand, (r?) = jb2, where b is a segment length. So, (rjJ2)= [(x -j>j/x]b2. The ends in question must, after all, be found somewhere from r = 0 to a, which yields
y , ( r ) * (~2/2*(r,,2))3/2ex~(-3~/2(r,,2)) (36) According to Kuhn's insight,16 and Appendix I1 in the preceding paper,' we find Pj/K
wj.m = wPjJ(0) = ( 3 / [ 2 ~ ( x- j ) j / ~ ] ) ~ / ~ 0 (KR,KR,,/K)/KR, for ideal chains
= Px-j/K =
(37)
This relationship was fmt found by Zimm and S t ~ c k m a y ein r~~ the derivation of the mean radius of p t i o n of a cyclic molecule, being later generalized by Fixmad to involve excluded volume effects. The physical meaning of eq 37 is simple: " h e probability of internal /-ring formation is equivalent to the probability of
The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9501
Interchange Reactions during Polymerization occutrence of thej-ring formation under the condition of forming a loop between the ends of an x-chain." This may be expressed with Fixman's notations as P(rj=OJring) = P(rj=O,ring) /P(ring) 1-5. Transformation of State to Time. As referred in the preceding report,' A-' = r(i) has the dimension of time and the kinetic forms represent general nth-order kinetic equations. Hence, the time that it takes to reach state k must have the form
1.0
10-1
10-2 Nxl L Nx
10-3
where Ncjfil denotes the molecule number of a component, cj, as a catalyst and cj an exponent of cj related to reaction order. II. Discussion Ip-1. Equilibrium Condition. We first seek a condition of simultaneous equilibrium between the polycondensation and the interchanges, and next, following the result obtained, we show numerical examples of the presented theory. An important question is whether an equilibrium state can be realized in the system under discussion, i.e., whether or not it is possible that the equilibrium point of the polycondensation agrees with those of the U-B and the B-B interchange. With the help of the combinatorial argument in section I, we can answer this question. Consider a simple case where every reaction is elementary. Let us note a j-ring in equilibrium. The principle of detailed balance requires an equilibrium between all the components, so that (a) for the polycondensation
10-4 0
0.2
0.4
0.6
0.8
1.0
Extent of Reaction:D
N~xlNo
kLNj,kNx,k/ v = kBNEkNj+x,k/ kRpj,k = k$v,jnr,,/
(39)
(b) for the U-B interchange KuNxJNR,,k/
v = KURpj+x.k
(40)
(c) for the B-B interchange K(x - )NxJNR,,k/ dNR#NRmk/
v = KR,(x
v = Pjo'
- l)Nj+x,k x)NRpmk
(41) (42)
The question is whether these equations have a common solution: eq 39 quickly yields the Stockmayer result16 [RjlD(=NRbk/
v) = (kR,/ kL)D*'/j
(43)
where D* is the extent of reaction of acyclic species alone. From this information and eqs 40 and 41, one finds a necessary condition for eqs 39-41 to have a common solution kR,/kL = K'R,/K"
= KR,/K
(44)
On the other hand, the substitution of eqs 43 and 44 into eq 42 leads us to the relation (35) Pj
. . . . . . . . . . . 0
= KR,KR./KR,+.
Therefore, the above equations involving (42) have a common solution only if relation 44 is fulfilled. Hence, we have obtained a necessary and sufficient condition (44) for the simultaneous equilibrium between the polycondensation and the interchanges. 11-2. Distortion of Mdecular Distributioa. We want to calculate the main characteristic of the present solution. For this purpose, we choose an extreme case where the U-B or the B-B interchange predominantly occurs. In Figure 2 is illustrated such a case: Let a polymerization (denoted by +) commence from chain monomer and a depolymerization (-) from the ring monomer. Dotted lines denote the Flory distribution of acyclic species (Figure 2a) and the Stockmayer distribution of cyclic species (Figure 2b). We recall that the ideal distributions of chains and rings have worked as a good approximation in nonequilibrium system with ring formation,' so that they have more universal character than supposed from the restrictions imposed on their theories.
0.2
0.4
0.6
0.8
1.0
Extent of Reaction: D
Figure 2. Trajectories of chains (a, top) and rings (b, bottom) plotted as functions of the extent of reaction for a case where the interchanges predominantly occur. Chains are assumed to be Gaussian. Initial conditions: No = 1O00, C,= 1.0, kL = kB = 1, K = K" = 10, kR,/kL = K"R /K" = KR,/K = ( 3 / 2 ~ j ) 3 1 2 (i) ; for the polymerization Nl,o= 1O00, = ho = Q = 0; (ii) for the depolymerization (+) NR, = 1O00, NQ = ho 1o00, &) = 0. (4)
&,
With this universality in mind, we find that the interchanges have made nontrivial deviation from the ideal distributions (Figure 2). Note that such differences can never be observed in the absence of the interchange reactions. This may be interpreted as a consequence of strong antagonistic flow toward two entirely different distributions; Le., the geometrical distribution and the binomial one (see the following section 11-3). This is an essential feature of the present solution. In the following application, we see this extreme example in the familiar Nylon-6 polymerization. 11-3. Ring-Opening Polymerization. We notice at once that the solution reduces to the known ring-opening polymerization if we restrict our solution to the U-B interchange. Let the reaction commence with cyclic monomer and a small amount of acyclic monomer as an initiator. No ring formation is taken into account here: (1) First we permit the U-B interchange of all bonds including acyclic species; of course, the counterparts are terminal f.u. Not surprisingly, the distribution of acyclic species asymp totically approaches the most probable one with increasing state or time (the graphic representation is omitted). (2) Secondly we permit the U-B interchange only for cyclic monomer. This type of reaction is called a ring-opening polymerization and found in anionic polymerizations of lactams and lactones.5a The resultant acyclic species distribute exactly in the binomial form throughout the entire process; Le., in state k NX,k/Nl,O = e-1)(1/W1(1 - 1/ak-*l (45) where Z = (No - hk) = C Ny,k(see Appendix 111for the mathematical proof).
Suematsu and Okamoto
9502 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 TABLE I: Equllibrim C ~ n r b (lux) ~ t ~in --chin Spoor and Zahn (525 K) Andrews, Jones, and Semlyen (525 K)"
average (525 K)
Equillbrin x = l
x=2
x=3
x=4
x=5
x=6
0.7430 0.7100 0.7250 0.5820
0.0350 0.0398 0.0374
0.0158 0.0200 0.0179
0.0129 0.0124 0.0127
0.0089 0.0076 0.0082
0.0042 0.0042
t t t t K)* t 'Rad from Figure 2 in ref 28. bCalculated by the equation K I ( T I )= Kl(T2) e x p ( - W AT/(RTIT2)),where W and T are the heat of cyclization of the cyclic monomer (x = 1) and temperature, respactively. Meggy reports A?P = 3560 cal/mol, while Tobolsky and Eisenberg report A?P = 4030 cal/mol. We here employ their average, AIP = 3800 cal/mol. (495
The same operation should not be performed for the quantities III. Application to the Polymerizatioa of e-Caprolactam such as (No- h) and h which express the number of u.f.u. In this section, the preceding theory is applied with a minor III-2. Rektionsbip betrrecnRate Consbats. To perform the amendment to the polymerization of e-caprolactam. numerical work of our theory, every rate constant must be proIn a review article by Imoto hued in 1960,19the polymerization vided. As one readily notices from the mathematical forms of mechanism of ccaprolactam has been introduced focusing on the the absolute values are not always works of Hoshino,20Mattes?I Kruissink:2 her man^,^' W i l ~ t h , ~ ~ the transition probabilities necessary. A key point is to find out relations among rate conand others.25 In summary, "pure e-caprolactam does not comstants, which is an essential feature of the present approach. mence polymerizing by heat, but commences by the addition of Fortunately, we can perform this through the simple argument water; the polymerization is accelerated by bases or acids; there of equilibrium. exists a latent time in the rate of the consumption of the lactam. First let us accept the following fundamental relations derived It had been long conjecturedmthat the initiation commences with in the preceding theory the hydrolytic ring-opening of the lactam into eaminocaproic acid, b e i i followed by the classical polycondensation between unreacted Pj = K R ~ R J K R ~ (35) functional units (u.f.u.)." Kruissink, Van der Wan, and Staverma# disclosed the imkRJ/kL =: K ~ R , / K =~ K R , / K (44) portance of the U-B interchange between e-caprolactam and e-aminocaproic acid. At the same time, Hermans, Heikens, and Note that eq 35 was derived based on ideal chain statistics, so van Velden analyzed the plymerizationz3initiated by water, and that it has no reflection of, say, excluded volume effects. Despite evaluated some rate constants with the help of differential this crude approximation, we may expect that eq 35 will work equations. Their results were in rough agreement with Wiloth's." for a real chain. Note that a cyclization rate constant is another Wiloth took account of the B-B interchange by introducing its expression of a cyclization probability. From this viewpoint, eq term into the differential equations which he integrated numer35 satisfies the mathematical form of the conditional probability. ically. The agreement with the experiments was fairly good for Hence, this may work irrespective of the chains chosen, only if a system with a small amount of water, but not for a system with the independent motion of the j chain and the (x - j) chain is a large amount of water. Tobolsky and EisenbergIz dealt with assumed. the same problem from an equilibrium viewpoint. [l] A relationship, kRx/kL,can be derived from the concenHowever, no one has 50 far addressed the effects of higher rings tration of cyclic species. For example, rearrangement of eq 39 and ring expansion. In what follows we shall reexamine all the yields possible mechanisms of the e-caprolactam polymerization from another viewpoint, taking account of all the effects neglected by k ~ , / k= ~gAR,I/D** E gdKx (49) previous workers. or equivalently III-1. A " n t of Theory. It is well-known that e-caprolactam (cyclic monomer: R1) is strained; Lylo H 3.8 kcalmol-' kR,/kL = gxxPW,m/D*x (50) according to Meggy'sZ6and Tobolsky-Eisenberg'slz contribution. So, we must start with a minor amendment to the theory. Folwhere K, is an equilibrium constant, p the density of a system lowing the principle of rate pnxxsses, it is natural that one assumes (not to be confused with the internal cyclization constant), w, the that the rate of bond dissociation is much faster for the cyclic weight fraction of x-ring, and m the mass of a repeatin unit. monomer in question. Hence, we introduce a ring-opening factor [2] By eq 44,kR,/kLis related to the other quantities, K R,/~u defined by g, = kB,R,/kB,lwhere g, is approximated to unity if and KR*/K, from which the internal cyclization rate constant pl x 1 2. One may regard g, as a statistical factor for a total bond can quickly be calculated. number in cyclic x-mer. As far as the dissociation of the cyclic [3] Still another relationship kL/kB can be derived as follows: monomer is concerned, the factor, gl, must be multiplied such that Let xW,o= Nw/No be the mole fraction of water in state 0. With NE,k= NEo hk - ho, we rearrange the first formula of eq 39 into the form
{a,
e
+
kL/kB
(xW,o+ D - Do)D*/(l - D)(1 - D*)
(51)
where Do denotes the extent of reaction in initial state. D and w, are experimentally determinable in equilibrium, so that D* is calculable from the known relationship D* = ( D - Cw,)/(l
- Cw,)
(52)
Hence, we have gained the important relationships among the rate constants. III-3. Ev8hmtion of Putmeters. kRx/kb This quantity is obtained from 50 with the weight fraction of x-rin following Spoor and Zahn Y 7 and Andrews, Jones, and Semlyen! Here we take their average (see Table I). kL!ks. With q 51 we can make a "rough" estimation of the quanttty kL/kB.The meaning of "rough" becomes clear later. The equilibrium extent of reaction D can be read from Figure 2A and
The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9503
Interchange Reactions during Polymerization
TABLE U C "
of the P 8 M h m C.lcuLtd Hermans et al. Wiloth this work 495
K
495
720.0
kLlkB kR,lkB
K
495
K
(525
K)
1640.0
833.3 5678.0
820.0 5488.3
(470.0) (2726.0)
4.1 123.0 EO
14-15 93.0 3.4
11.5 135.0 1
+ xzCSNXl3 =0 1
(A9
+ x 2 1SN,I4 = 0
(A61
Hence, the summation becomes
C6Nxm= 0
(A71
m
as asserted. Appendix II' Let Pi-l(x,L), Pi-l(x,R), Pi-,(x+j,B), and P,-I(Rx,B}be the probability that a f.u. on an x-mer chain propagates, the cyclization probability of a chain x-mer, the probability that a new chain x-mer is formed through the bond scission of the xth unit of an (x + j)-mer, and the probability that any one bond of a cyclic x-mer breaks in (i - 1) i, respectively. The respective variations of a cyclic x-mer and a chain x-mer numbers in (i - 1) i are then expressed as
-
6NRJ-l
-
= Pi-lIxN - P~-IIRx,BI
P I
CPi-lb&)Nx-j,i-l/(No
6Nx.i-l
j=1
(A81
- hi-1) +
X
2[ CPi-1b3)- CP,-I~,BII+ Pi-I{Rx,B) y= 1
Y=l
2Pi-l(~,L)- (X - 1) Pi-l(x,B) - Pi-l(x,RJ (A9)
The variation of a reacted f.u. number, ahi+ in all species and that, in acyclic species alone are respectively 6hi-l
=
C [Pi-l(X,L) + Pi-,{x,R)- j=CPi-I{x+j,B) - P,-I(RX,BI] I
x= 1
(A10) 6ti-l
= 6hi-1 - 6ZXNRI,i-I X'
1
...
(A1 1)
where Pi-l{x,LJ = (kLNx.i-1 CNy,i-l/V/A y-1
Pi-lhR) = kR,Nx,i-l/ A Pi-lb+j,Bi
e
(kBNE,i-INx+j,i-l/ v/A
P i - l k B ) = (kBNE,i-IXNR..i-l/ v / A
(A12)
9507
J. Phys. Chem. 1992, 96, 9507-9512 Nz, = N1,OkU - l/ak-'(l/a = NI,o(!)(~ - l/W1(l/a' By induction Nx,k
= Nl,O(kl)(l - l / a k - ' + ' ( l / a P 1
For both Z >> 1 and k Poisson distribution Nx,k/Nl,O
(4415)
>> x, this can be approximated to the (xP1/(x- I)!) exp(-x)
where A = (x) = k/Z. Registry No. c-Caprolactam, 105-60-2.
References and Notes (1) Suematsu, K.; Okamoto, T.Colloid Polym. Sci. 1992,270,405,421. (2) Suematsu, K.; Okamoto, T.J. Phys. SOC.Jpn. 1992, 61, 1539. (3) (a) Carothers, W. H. Chem. Reo. 1931,8,353. (b) Flory, P. J. J. Am. Chem. Soc. 1940,62, 2255. (c) Kotliar, A. M.J. Polym. Sci. 1981, 16, 367. (4) (a) Brubaker, M.
M.;Coffman, D. D.; McGrew, F. C. US Patent
2,339,237, 1944. (b) Beste, L. F.; Houtz, R. C. J. Polym. Sci. 1952.8, 395. (c) Han, M.J.; Kang, H. C.; Choi, K. B. Mucromolecules 1986, 19, 1649. (5) (a) Vert, M.;Chabot, F.; LeRay, J.; Chritel, P. Mukromol. Chem. Suppl. 1981,5,30. (b) Dayte, K. V.; Raje, H. M.J. Appl. Polym. Sci. 1985, 30,205. (c) Muhlebach, A.; Economy, J.; Johnson, R. D.; Karis, T.;Lyerla, J. Mucromolecules 1990. 23, 1803. (6) Carmichael, J. B.; Winger, R. J . Polym. Sci. 1965, 3, 971. (7) Stem, M. D.; Tobolsky, A. V. J . Chem. Phys. 1946,14, 93.
(8) (a) Grever, G. Mukromol. Chem. 1957, 22, 183. (b) Grever, G.; Oechmann, W. Mukromol. Chem. 1961,44228. (9) Catcheside, D. G.; Lea,D. E.; Thoday, J. M. J. Genet. 1945,47,113. (10) Alberts, B.; Bray, D.; Lewis, J.; Rail, M.; Roberts, K.; Watson, J. D. Moleculur Biolom of the Cell; Kyoikusha: Tokyo, 1985. (11) Flory, P. J. Chem. Rev. 1946,39, 137. (12) Tobolsky, A. V., Eisenberg, A. J. Am. Chem. SOC.1%9,8/, 2302: 1960,82,289. (13) Lertola, J. G. J. Polym. Sei. Part A,: Polym. Chem. 1990,28,2793. (14) Kondepudi, D. K.; Pojman, J. A.; Mansour, M.M.J. Phys. Chem. 1989, 93, 593 1. (1 5) Pojman, J. A.; Garcia, A. L.; Kondepudi, D. K.; Van den Bnxck, C. J. J. Phys. Chem. 1991,95, 5655. (16) (a) Kuhn, W. Kolloid 2.1934,68,2. (b) Jacobson, H.; Stockmaya, W . H. J. Chem. Phvs. 1950.18, 1600. (17) Zjmm, B. H.; Stockmayer, W. H., J . Chem. Phys. 1949, 17, 1301. (18) Fuman, M. J. Chem. Phys. 1955, 23, 1656. (19) lmoto, M.Kuguku 1960, 15, 540 (in Japanese). (20) Hoshino, K. Nihonkugukkuishi 1940, 61, 475 (in Japanese). (21) Mattes, A. Mukromol. Chem. 1951, 5, 197; 1954, 13, 90. (22) Kruissink, C. A.; Van der Want, G. M.; Staverman, A. J. J . Polym. Sci. 1958, 30, 67. (23) Hermans, P. H.; Heikens, D.; Van Velden, P. F. J. Polym. Sci. 1958, 30, 81. (24) (a) Wiloth, F. Kolloid 2.1955, 144, 58. (b) Wiloth, F. Kolloid 2. 1958,160,48. (c) Wdoth, F. Z . Phys. Chem. N.F., 1%7,11,78. (d) Wiloth, F. Mukromol. Chem. 1959.30, 189. (25) Ogata, N. Mukromol. Chem. 1959,30, 212. (26) Meggy, A. B. J. Chem. Soc. 1953,796. (27) Spoor, H.; Zahn, H. 2.A w l . Chem.1959,168, 190. (28) Andrews, J. M.;Jones, F. R.; Semlyen, J. A. Polymer 1974, IS, 420. (29) See The Merck Index; Merck & Co.: Inst. 1976; p 59.
Thermodynamic Analysis of Scanning Calorimetric Transitions Observed for Dilute Aqueous Solutions of ABA Block Copolymers N. M. Mitchard, A. E. Beezer,' J. C. Mitchell, J. K.Armstrong? B. Z. Chowdhry,+ S. Leharne? and G. Bucktod Chemical Laboratory, The University of Kent at Canterbury, Canterbury, Kent, CT2 7NH, U.K, (Received: April 8, 1992; In Final Form: July 18, 1992)
Dilute aqueous solutions of a series of poly(oxyethylene)-poly(oxypropylene)-poly(oxyethylene) block polymers have been shown to undergo phase transitions. A high-sensitivity differential scanning calorimetric (HSDSC) study of these block copolymers has enabled the thermodynamic parameters for the phase transitions to be obtained. The thermodynamic parametera are all dependent on poly(oxypropy1ene) content of the polymer and not on poly(oxyethy1ene) content or total polymer molecular mass. The importance of the poly(oxypropy1ene) in determining the aqueous solution phase properties of these polymers is further emphasized by the positive value for the heat capacity change from pre- to postphase transition. A positive heat capacity change is characteristic of "melting" of water ordered by exposad nonpolar groups. The work reported in this paper shows that the thermodynamic parameters describing the polymer phase transitions can be derived from basic thennodynamic principles and from consideration of the poly(oxypropy1ene) content. The theoretical analysis described in this paper also predicts that these polymers should undergo a second phase transition at high temperature and indeed these transitions have been detected by HSDSC.
Iatroduction
Previous work by Rassing et al.5 and Gilbert et a1.6 and our own investigation of P237 (P237 consists of 70% ethylene oxide (Eta) and 304%Propylene Oxide (pro) (p237; aveWetotal molar M1.t 7700 of which MI for E t 0 = 5390 and Mr for Pro = 2310)) by NMR3 show that the temperature associated with the phase transition is related to the properties of the hydrophobic portion of the molecule. The observed positive heat capacity H(OCH~-CH~)a-(OCHMeCH~)~-(O-CH2-CH2)oOH chanae for the ore- to DostDhaSC transition M related to. as noted by P&lov,8 "&e gradA melting of water ordered by the expo~urc and therefore have distinct hydrophilic and hydrophobic regions. Of non-polar groUP***" (see refs 7-1 By altering the size and ratio of the two regions it is possible to It was suggestedlZ that the thermodynamic parameters dealter the solution properties of the polymers and so to affect the scribing the phase transitions could be fitted to equations of the physicochemical characteristics of the phase transitions. form X,, = aMr(Pro) bMr(Eto) (1) iSchool of Biological and Chemical Sciences, Greenwich University, London SEI8 6PF, UK. wherex is a thermodynamic f U l W h Such as enthalpy Or e n m y tSchool of Pharmacy, University of London, Brunswick Square, London WClN 1AX. UK. change and a and 6 are constants. This functional form requires
I,,
p ~ b ~ ~ t i o n sit1was -4 that in dilute solutions of poloxamers, poly(oxyethy1ene)-poly(oxypropy1ene)-poly(oxyethylene), ABA block copolymers undergo reversible phase transitions. The poloxamers have the general formula
+
0022-3654/92/2096-9507$03.00/00 1992 American Chemical Society
