SECOND OSMOTIC VIRIALCOEFFICIENT OF ATHERMAL POLYMER SOLUTIONS
3647
On the Second Osmotic Virial Coefficient of Athermal Polymer Solutions
by G. V. Schulz, H. Baumann, and R. Darskus Institut f a r physikalische Chemie, Universitlit Mainz, M a i m , Germany
(Received July I , 1966)
+
+
In the usual equation for the osmotic pressure, R / C = RT(M-' Azc . . .), the second virial coefficient, A z , can be split up into an enthalpy and an entropy term, A t , , and Az,s. For athermal solutions, defined by Az,H = 0, many statistical equations have been derived which have since been abandoned for two reasons: they yielded too high values for Az and they could not explain the decrease of A z with increasing molecular weight. As is known today, the failure of these theories is caused by neglect of the fact that the concentration of segments near another segment is different from the average segment concentration in the solution. This effect is considered by the h(z) function in the newer theories. Experiments in many different systems make it appear probable that for good solvents the h(z) function of Casassa is the best approximation. Since solvents giving athermal systems are good ones, it is proposed to combine the previously derived equation (7) for athermal solutions with the Casassa function. Light-scattering measurements in two approximately athermal systems (polystyrene in toluene and poly(methy1 methacrylate) in tetrahydrofuran) show that it is possible in this way to calculate Az and its dependence on molecular weight without using an adjustable parameter.
I. Introduction. Pseudoideal and Athermal Solutions In early statistical-thermodynamic treatments of polymer solutions, most attention was directed to athermal solutions, defined by the condition
AH1 = 0 (1) (AH1 = heat of dilution). For such solutions one may expect that the thermodynamic behavior is governed only by the statistics of geometrical configurations of the dissolved molecules and therefore can be calculated neglecting energetic interactions. However, for some time now, interest has been turned away from the theories based on the assumption of eq 1, because none of them could explain the generally observed decrease of the second osmotic virial coefficient with increasing molecular weight of the dissolved polymer. Later, many efforts were made to solve the problems of polymer solutions using the cluster theory of Mchiillan and Mayer.1 All equations derived on this basis contain the excluded volume p in the same form as in the theory of real gases
and a function h(z). characteristic of the conformations of coiled long-chain molecules. Hereby, solutions in the vicinity of Flory's 8 temperature, at which the second virial coefficient A z (as well as the excluded volume 0)vanishes, became the focus both of theoretical considerations and of experiments. For solutions in which the equation
(3) but not eq 1 is valid, the name pseudoideal solutions was proposed.2 In fact, all known 8 solutions are not ideal but pseudoideal, because they have substantial positive values of AHI. Many equations for the function h(z) have been proposed. Concurrently, many experimental investigations on the osmotic behavior of polymer solutions have shown such a diversity of behavior that it seems possible to find one or more systems to fit each h(z) function proposed. Conversely, it seems to be impossible to find an h(z) function of general validity. The source of the difficulty may be that pseudoideal A2 = 0
(1) W. G. McMillan and J. E. Mayer, J . Chem. Phys., 13, 276 (1945). (2) G.V. Schulz and H.-J. Cnntow, Z . Elektrochem., 60,517 (1956).
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Nooember 1966
3648
G. V. SCHULZ, H. BAUMANN, AND R. DARSKUS
solutions are thermodynamically rather complicated.
If the ideal solution is defined by eq 1 for the enthalpy of dilution and by eq 4 for the entropy of dilution ASlid = - R In (1 - x2)
(4)
(xZ = mole fraction of the solute), one can see immediately that the athermal solution deviates from the ideal one only by the nonvalidity of eq 4, but eq 1 is maintained. However, for pseudoideal solutions, defined by eq 3, both ideality conditions (1) and (4) are not realized. If one divides the second virial coefficient into an entropy and an enthalpy term Az
=
Az,s
+
- 4 2 , ~
(5)
it follows that condition (3) is realized by cancellation of the two nonideal t’erms. Thus the pseudoideal solution is defined by the equation
Az,s
=
-Az,H
(6)
It has been shown experimentally that at the e temp e r a t ~ r e , ~the - ~ heat and the excess entropy of dilution have extremely high values. These considerations suggest that further study of athermal solutions might be promising. They are closer to ideal ones, since they fulfil the ideality condition (1). hIoreover, experiments in different solvents have shown that the lowest values of Az,s are obtained when A z , , is zero, although A2 in this case is relatively high.6 In Figure 1,’ this effect is represented for poly(methy1 methacrylate) in 15 pure solvents. Tho scatter of the osmotically determined points seerns to correspond more to experimental errors than to real deviations from the general curve. In section 11, we propose a method of combining the older theory of athermal solutions with the newer results of statistical calculations. In section 111, experiment:, carried out to test these assumptions are reported. Some experimental details are given in section V. A difficulty arises from the fact that in contrast to e solutions, which can be realized exactly since for many solvents the 8 temperature lies in an easily accessible regicn, athermal solutions are rare, because their realization depends on conditions which are it is removed from Our influence’ only to find approzimately athermal SOlUtions, which may be defined by the relation A2,, (( A z , This i,3 satisfied by solutions of polystyrene in toluene, and less well by poly(methy1 methacVlate) in tetrahy(-trofuran. However, the result is satisfactory in both cases. The Journal oj’ Physical Chemistry
-
A2,,’(cm3mole.g4)
Figure 1. Entropy term A z , as ~ a function of the enthalpy term A 2 , xfor poly(methy1 methacrylates) in 15 solvents: - - - -, averaged curve for Az; 0, light-scattering measurements ( M = 210,0007 and 198,000 (this work)); 0, osmotic measurements3 (144 = 129,000).
11. Theoretical Considerations From the different equations derived by several authors for the second virial coefficient of athermal solutions,s we use9
(7) where fisp is the partial specific volume of the polymer, M,,, is the molecular weight of the monomer unit, a is its length in the chain (2.54 A for vinyl polymers), and d is the averaged diameter of the worm-like chain molecule, which can be calculated by the relation N A ( s / 4 ) d 2 a = O s p M m o n with the result AZath = (Oss/Mmon) (7ra/4d)
aa/4d = (7r3a3N~)1’2(43n~,onfisp) -’” (8)
is Avogadro’s number. Equation 7 is derived by statistical geometrical calculations assuming a continuous medium. Similar equations are given by Zimm’O and Huggins. l 1 On the basis of the lattice model, Miller,’2 Guggen-
N A
(3) G. V. Schulz and H. Doll, Z. Elektrochem., 56, 248 (1952); 57, 841 (1953). (4) G. V. Schulz and H. Hellfritz, ibid., 57, 835 (1953). (5) G. V. Schulz, H . Inagaki, and R. Kirste, Z . Physik. Chem. (Frankfurt), 24, 390 (1960). (6) G. V. Schulz, A. Haug,and R. Kirste, ibid , 38, 1 (1963). (7) R Kirste and G. V Schulz, ibid., 27, 301 (1961). (8) A detailed and critical synopsis of all these statistical theories of athermal solutions is given by A. Manster in “Physik der Hochpolymeren.” vel. 11, H. A. Stuart, Ed., Springer, 1953, Chapter 11. l ~ + q i ~ ~ uZ l. Ndurfor8ch.7 z 3 2a, 27 (1947); Makromol. Chem.,
T,)
(10) B. H. Zimm, J . Chem. Phys., 14, 164 (1946).
(11) M.L. Huggins, J . Phys. Colloid Chem., 52,248 (1948). (12) A. R. Miller, Proc. Cambridge Phil. SOC.,39, 54, 151 (1943).
SECOND OSMOTICVIRIALCOEFFICIENT OF ATHERMAL POLYMER SOLUTIONS
3649
heim, l 3 Miinster, l 4 and others arrived a t equations which can generally be brought to the forms
(15)
~z~~~
=
(gsp/Mmon)(n
- 2)/2n
(9)
where n is the coordination number. All of these equations, although of different mathematical forms, are not very different in the numerical values for A2. From eq 7 and 8, using known values of gSp, M m o n , and a, one finds for polystyrene Azath = 20 X and for poly(methy1 methacrylate) AZath= 19.2 X cm3 mole g-2. However, all experiments give considerably lower values for AB, which decreases with increasing molecular weight of the polymer (see Figure 3). For low molecular weights, corresponding to one statistical chain element, the experimental values of A 2 approach the theoretical ones.'5~'~ These differences between experiment and theory are due to the fact that the concentration of segments in the neighborhood of a segment is not identical with the average concentration of segments in the solution, as first pointed out by F1ory.l' For the following discussion, we use the general equation of Zimm, Fixman, and StockmayerlB
Azath = (g~p/Mmon)(s~/4d)h(~)
If, moreover, an equation of general validity would exist for h(z), the old attempt of deriving the second virial coefficient of athermal solutions on a purely geometrical basis could be successful. With this aim, we may proceed as follows. Comparing eq 13 with eq 15 we obtain
BO = 2Mo2(gBp/Mmon)(*a/4d) From the definition of N it follows that
N = M/Mo
x = N "2/3 (3/2 nb2)a'2
(11)
is a parameter characteristic for solutions of coiled long-chain molecules consisting of N chain elements of length b and molecular weight Mo. Introducing Flory's 0 temperature, one can simplify eq 2 to
B
= Po(1
- 0/T)
(12)
Recent,ly, KirsteIg has shown that for very different p(z) functions, eq 12 is a good approximation in a considerable range of temperatures not too far above 0, p(x) being regarded as independent of T. Assuming that for an atherma' = 'OK, one obtains from eq 10 and 12, setting
NAP
B; NAP0
=
Bo
AZath = (Bo/2M02)h(2)
(13)
hence
zath = N'''(BO/NA>(3/2ab2) "'
(14) effect Since one can suppose that the represented by the function h(z) in eq 10 exists also in athermal solutions, one can extend eq 7 to
(17)
With the aid of the equation
6gz
=
Nb2
(18) in which is the mean square of the unperturbed radius of gyration, the length of the statistical chain segment is found from (19) So we obtain, by combination of eq 14 with eq 17, 16, 8, and 19 b2 =
$th
where the function h(x) contains the above configurational effect and
(16)
6 M 0 ~ ~ 2 / 1 M
= 1/32(%p/N,k) '/'(a/Mmon)
" ' M ' / ' ( ~ / M-)"'
(20)
All values on the right-hand side of eq 20 are known or can be determined by light-scattering measurements in a 0 solvent. Some uncertainty could arise from the possibility that a t the 0 points the solvents may influence the "unperturbed" dimensions. 20-z2 To reduce this uncertainty, it is useful to carry out measurements in several 0 solvents the 0 points of which lie in the temperature range where the investigation of the athermal solution is made (compare section 111). Moreover, recentlylZ3 Baumann has shown by an ext'rapolation method using Fixman's equation for the expansion factor that there are negligible solvent effects on the unperturbed dimensions of polystyrene and poly(methy1 methacrylate) in the temperature range of this investigation. (13) E. A. Guggenheim, Proc. Roy. SOC.(London), A183, 203 (1944). (14) A. Monster, Kolloid-Z., 105, 1 (1943); Makromol. Chem., 2, 227 (1948); 2. hraturforsch., 1, 311 (1946). (15) G. V. Schulz and G. Meyerhoff, 2. Elektrochem., 56, 545 (1953). (16) G. V. Schulz and H. Marzolph, ibid., 58, 211 (1954). (17) P. J . Flow, J . Chem. Phys., 17, 1347 (1949). (18) B. H. Zimm, M. Fixman, and W. H. Stockmayer, ibid., 21,1716 (1953). (19) R. Kirste, 2. Physik. Chem. (Frankfurt), 39, 20 (1963). (20) K. J. Ivin, H. A. Ende, and G. Meyerhoff, Polymer, 3 , 129 (1962). (21) W. Burchard, Makromol. Chem., 56, 239 (1962). (22) G. V. schuiz and H. Baumann, ibid., 6 0 , 120 (1963). (23) H. Baumann, J. Polymer Sci., B3, 1069 (1965).
Volume '70,Number 11 -Vovember 1966
3650
G. V. SCHULZ, H. BAUMANN, AND R. DARSKUS
A more severe problem is the selection of the h(z) function, because of the great number of different equations which have been proposed. Most of the calculated curves6 lie between two limiting curves, the flattest of which is that of Orofino and F l ~ r y , which ~* numerically corresponds almost exactly to that of Isihara and Koyama.26 The other limiting curve is given by the equation of Casassa,26which deviates a little from the older equation of Casassa and Markowitz.27 Measurements in a number of different systems (three series of polymers and 11 solvents) carried out in our laboratory6,2H make it seem probable that the h(z) function depends on the “goodness” of the solvent. For poor solvents (ie., near the 8 point) the OrofinoFlory function seems to be a good approximation. With increasing solvent power, a transition to the Casassa function takes place, and for good solvents the dependence of A2 on molecular weight is well described by this function. Since athermal solvents give a relatively high A2 value, they can be regarded as good solvents, so it is probable that this function can be applied to athermal solutions. CasassaZ6has derived the equation
A
B
6
%
3 2*? 3.
‘ 1
5 4
3 2 1
10
20
30
40
50
Figure 2. Second virial coefficient vs. temperature measured by light scattering for two approximately atheimal systems: A, polystyrenasin toluene; B, poly(methyl methacmlata) in tetrahvdrofuran,
tetrahydrofuran are represented. The results for the system PRIMA in T H F are less precise, possibly because of the lower refractive index increment and the difficulty of purifying the soivent. The results of the measurements are given in Table I. The values of A2,, for the polystyrene solutions are very low compared with A2. Therefore this system is, to a good approximation, an athermal solution. h(z) = [l - exp(-5.68~~~~-~)]/5.68~~~~-~ For PMMA the values of A 2 , H are negative, showing a (21) positive heat of dilution and a considerable scatter, aZ5- aZ3= 2.0432 but no trend of the values with molecular weight is If we accept this equation and combine it with zath evident. It is therefore permissible to use the average from eq 20, we can calculate Azath using eq 15 and 8 cm3 mole g-2. According value of A 2 . H = 1 X on the basis of simple geometrical assumptions and to the definition of the approximately athermal solulight-scattering measurements of the unperturbed tion given in section I, solutions of PMRIA in tetradimensions under e conditions. hydrofuran are not very far from being athermal. In the next section, we will try to verify the above For the application of eq 21 and 20, the ratio r g , , 2 / ideas by light-scattering measurements in two quite M , is required. This was determined for polystyrenes different systems: polystyrenes in toluene as a nonin two e solvents by light scattering measurements. polar system and poly(methy1 methacrylates) in tetraThe result is shown in Table 11. The average for three hydrofuran as a polar system. Preliminary measurefractionated samples of PS in two solvents is ments have shown that these two systems are nearly M , = 7.75 X lo-’* cm2 mole g-l. For PRIMA the athermal ones. radius of gyration was previously determined in the 8 solvent butyl The average value is III. Experimental Results for the Systems M , = 4.8 X lo-’* cm2mole g-l. Polystyrene (PS) in Toluene and Poly(methy1 In Table 111, the Az values for athermal solutions as methacrylate) (PMMA) in Tetrahydrofuran predicted by eq 15 with the help of eq 8, 20, and 21 are From the slope of the plot of A 2 us. T, one can calculisted and compared with the values found by experilate the entropy and the enthalpy components of A2 (24) T. A. O r o h o and P. J. Flory, J. Chem. Phys., 26, 1067 (1957). with the equation5
s/ gz/
A z , , = T(bAz/bT) Az,H =
+ Az(1 - aT)
T[aAz - (bAz/bT)l
(22) (23)
where LY is the thermal expansion coefficient of the solution, or in dilute solutions of the solvent. I n Figure 2, the measurements in two systems in toluene and in The Journal of Physical Chemistry
(25) A. Isihara and R. Koyama, ibid., 25, 712 (1956). (26) E. F. Casassa, ibid., 31, 800 (1959). (27) E. F. Casassa and H. Markowits, ibid.,29, 493 (1958). (28) H . Baumann, R. Darskus, and G . V. Schuls, Communication t o the International Symposium on Macromolecules of the IUPAC, Prague, Aug-Sept 1965. (29) R. Kirste and G . V. Schulz, Z. Physik. Chem. (Frankfurt), 30, 171 (1961).
SECOFD OSMOTICVIRIALCOEFFICIENT OF ATHERMAL POLYMER SOLUTIONS
3651
Table I : Entropy and Enthalpy Components of the Second Virial CDefficient,of Approximately Athermal Solutions MLS X 103
A~ x 104, cma mole
A2.s X 10'. cma mole 8-2 (eq 22)
BA,/bT X 107
g-2
A z , x X lo4. cma mole g-2 (eq 23)
Polystyrene in toluene at 40" ( a = 1.12 X 209 4.28 4.8 4.28 0.0 570 3.30 2.7 3.0 0.3 1600 2.46 2.2 2.26 0.2 Poly( methyl methacrylate) in tetrahydrofuran at 25" ( a = 1.10 x 10-3) 7.1 5 . 3 12.0 47 6.0 4.8 69 4 5 -1.2 4.45 3.25 7.6 198 -1.3 = L o 470 2.65 7.4 3.95 1670 1.85 7.0 3.35 -1.51 9300 1.1 1.6 1.2 -0.1 J
Table 11: Relation between the Weight Average of the Square of the Radius of Gyration 2 and Molecular Weight of Polystyrene in Two 0 Solvents
r
-(e
Hexyl-m-xylene
,-(e
= 13O)--
Diethyl malonate = 3z.50)(7e"/MW)
( S / M W )
MLS X
X 108,
cmp mole
X lOS,
x 1018, cm*mole
10-8
cm
g -1
cm
g -1
570 1600 6000
218 350 672
212 357 658
7.9 8.0 7.2
( 3 ) ' / 2
x
1018,
8.3 7.7 7.5 Average 7.8
(G2)'/2
7.7
Table 111: Calculated Values of Apth for Two Systems in Comparison with Experimental Values (A2 in cm3 mole g-a) Mw X
eth
10-3
(e9 20)
h(z) (eq 21)
-M
Figure 3. Second virial coefficient A2 va. molecular weight for two approximately athermal systems. The curves are calculated by eq 15 with the help of eq 8, 20, and 21. A. Polystyrene in toluene. Osmotic measurements: +, Mark and Frank;31 X, Bawn, Freeman, and Kamaliddin3* and Bawn and Wajid;aa [XI, Krigbaum and Flory;s4 A, Schulz and Hellfritz;4 0, Stein;3s , light-scattering measurements: 0, Outer, Carr, and Zimm;as 0 , this work. B. Poly(methy1 methacrylate) in tetrahydrofuran. Osmotic measurement: +, Schulz and Doll; light scattering: 0, this work.
A$" X 104 (eq 15)
A2 X 104,
exptl
PS in toluene at 40' (nap = 0.92 cm*/g; M m o n = 104; a = 2.54 X cm) 209 3.16 0.228 4.56 4.28 570 5.21 0.178 3.56 3.30 1600 8.72 0.140 2.80 2.46 3920 13.67 0.114 2.28 1.84 PMMA in T H F at 25" (nep = 0.81 cma/g; M,,, a = 2.54 X 10+ cm) 25.7 2.20 0.269 5.16 47 2.97 0.234 4.50 69 3.60 0.212 4.07 198 6.10 0.166 3.19 470 9.40 0.136 2.61 1670 17.7 0.101 1.94 9300 41.8 0.068 1.31
=
100; 4.7 5.28 4.52 3.23 2.63 1.83 1.1
ment. For dBP) we have used the result of the measurements with H~ffmann.~O In Figure 3, the curves are calculated as described above for athermal solutions and the points are experimental values. In the system PS in toluene, many measurements of different authors made in the last 15 years are a ~ a i l a b l e . ~ ' -In ~ ~general, one can see that for the solutions in toluene the measurements scatter around the theoretical curve by f15% (on the average). A deviation of this magnitude is reasonable considering the experimental errors, including the uncertainty in the evaluation of A z from experiments. The measurements in the system PMMA in T H F are in better agreement with the theory than could be expected. This result shows that the treatment of athermal solutions discussed in section I1 gives a satisfactory approximation for the systems investigated, especially (30) G . V. Schulz and M. Hoffmann, Makromol. Chem., 2 3 , 220 (1957). (31) H. P. Frank and H. F. Mark, J . Polymer Sci., 6 , 243 (1951). (32) C. E. H. Bawn, R. F. J. Freeman, and A. R. Kamaliddin, Trans. Faraday Soc., 46, 862 (1950). (33) C. E. H. Bawn and M. A. Wajid, J . Polymer Sei.. 12, 109 (1954). (34) W. R. Krigbaum and P. J. Flory, J . A m . Chem. SOC.,7 5 , 1775 (1953). (35) D. J. Stein, unpublished measurements. (36) P. Outer, C. I. Carr, and B. H. Zimm, J . Chem. Phys., 18, 830 (1950).
Volume 70,Number 1 1
November 1966
G. V. SCHULZ, H.BAUMANN, AND R.DARSKUS
3652
if one considers that no adjustable parameter is used in the equations. Of course, it would be very useful to find more (approximately) athermal solutions and to test the general validity of the system of equations proposed. I
BI
82
IV. Discussion It is of interest to compare the temperature-independent part, PO, of the excluded volume of endothermic solutions (e solutions) with the value proposed for atherma1 solution. For e solutions, one may write?
endothermal
alher mal
(82 dpproximdtely)
P ~ / = M (~ 2~ ~
~
4
(24)
The quotient Po/Mo2 is thus accessible from purely thermodynamic data. For athermal solutions, comparison of eq 10 and 15 gives Po/Mo2= (2,”~)
(~sp/Mmon)
( ~ ~ / 4 d ) (25)
in which nn/4d is given by eq 8. The right-hand side of eq 25 contains only parameters of the structure of the macromolecule. For our understanding of the solution properties, it is important to investigate the relation between the values of PO/Mo2calculated from eq 24 and 25. The parameter Po in eq 12 is determined by the potential function cp(x) of eq 2. Introducing a Sutherland potential, Kurata and Y a m a k a ~ found a~~
p
=
(4T~03/3)(i- e / T )
that is
Po
=
4axo3/3
(26)
The significance of xo is shown for a number of potential functions in Figure 4. The hard-sphere potential A2 for the athermal solution corresponds to the Sutherland potenlial A1 for the endothermic solution. Equation 26 results from both these potentials. If a “hard” repulsion potential is operative [cp(x) = a for x < xo], the ratios Bo/Mozfor athermal and 8 solutions become identical. (In eq 2, a spherically symmetric field of force is assumed. This condition is not satisfied for polymer molecules, but it may be assumed as a first approximation that the thread-like nature of the molecules can be taken into account by a general correction factor to eq 2.) Values of Po/Mo2 calculated according to eq 25 for polystyrene and poly(methy1 methacrylate) are compared in Table IV with those obtained using eq 24 from previously published’v22 thermodynamic data in 8 solutions. The agreement in the order of magnitude evident in the table is remarkable, since very different solution states are being compared and the data required for eq The Journal o,f Physical Chemistry
Figure 4. Different potential functions. A1 (Sut.herland potential) and A2 “hard” potential functions; B1 and B2 are more general forms of the potential for endothermal and (approximately) athermal solutions.
Table IV : Comparison of the TemperatureIndependent Part of the Excluded Volume in e Solutions and in Athermal Solutions S o h tion
Polystyrene Athermal, eq 25 n-Butyl formate Hexyl-m-xylene Decalin Diethyl malonate Cyclohexane Diethyl oxalate Methylcyclohexane Cyclohexanol
e, “ C
( B ~ / Mx~ 10s’ ~ ) (eq 24 and 25)
(a,= 170,00022) ... -9 12.5 29.5 31 34 51.5 68 83.5
6.6 5.0 4.5 5.8 2.5 10.2 3.25 7.3 10.6
Poly(methy1 methacrylate) ( lgw= 210,0007) Athermal, eq 25 ... 6.5 I-Clilorobutane 36 3.3 4Heptanone 40.4 2.6 Isoamyl acetate 57.5 2.3
24 and 25 are quite different. The scatter of the values in 8 solutions may have a number of causes, including variations in Po and M o and to some extent experimental errors. A change in conformation of polystyrene may be responsible for the high value of Po/Mo2 in cyclohexanoLa8 I n general, of course, a “hard” cp(z) function cannot be assumed. A better approximation is a LennardJones potential. According to S t ~ c k m a y e rpotential ,~~ functions containing several minima cannot be excluded (37) M. Kurata and H. Yamakawa, J. Chem. Phys., 2 9 , 311 (1958). (38) C. Reiss and H. Benoit, Compt. Rend., 253, 268 (1961). (39) W.H.Stockmayer, Makromol. Chem., 3 5 , 54 (1960).
SECOND OSMOTIC V I R I A L C O E F F I C I E N T O F
ATHERMAL POLYMER
3653
SOLUTIONS
30
9 -2 0 0
E
Y
. I
P
h
3 a \
x“
10 10
f
a 6
0
4 0
4
1
2
3
4
5
7
6
Figure 5 . Zimm plot for polystyrene S114 in toluene a t 25” and X 4360 A ; 0 , plot of (Kc/R,y=o)’’’ VS. C.
for solutions (B1 and B2 in Figure 4). The simple relation (26) is not generally valid for such functions. Thus, according to Kirste,lg for Lennard-Jones 12-6 and 8-4 potentials, Po = 0.8 X 4xzo8/3 and 0.43 X 4azo3/3, respectively. For simple and multiple (‘square-well’’ pot,entials, the deviations from eq 26 are slight. In such comparisons it must be kept in mind, however, that the function p(x) derived for gases is of only limited applicability to solutions, since the space between the polymer segments is filled with solvent molecules. Further, p(z) depends also on the angle of approach of the chains containing the two interacting segments. Therefore, p(z) is a complicated average. lloreover, the assumption that p(z) does not depend on temperature is of doubtful validity, if only on account of the thermal expansion of the solvent. I n view of these difficulties, better agreement between theory and experiment at present probably cannot be expected.
and Scheibling (Sofica, Paris). The incident light (wavelength 4360 A) was vertically polarized. Refractive index increments were determined at the same wavelength with the differential refractometer of Schulz, Bodmann, and Cantow40 modified as described by B ~ d m a n n . Calibration ~~ of the scattering intensity and purification of solvents and solutions were described previously.’ The data were treated by the method of Zimm. Extrapolation to zero angle could be made linearly from low angles. The lines for the concentration dependence, however, were curved. Therefore, the molecular weights and second virial coefficients were ’ / ~c in analogy to obtained from plots of ( K C / R , = ~ )vs. the plot proposed by Flory for osmotic measurements. In this way, straight lines of intercept LW‘’~ and slope A , were obtained. The molecular weights agreed well with the corresponding ones determined in 8 solvents. (See Figure 5 . ) Two of the samples used were the ‘(international” polystyrene fractions S l l l and 5114
V. Experimental Section The light-scattering measurements were carried out with the commercially available photometer of Wippler
(40) G.V. Schulz, 0. Bodmann, and H.-J. Cantow, 2. Yaturforsch., 7 a , 760 (1952).
(41) 0.Bodmann, Chem. 1ngr.-Tech., 29, 486 (1957).
Volume 70, Number 1 1
Noiember 1966
B. S. AL-NAIMY,P. N. MOORTHY, AND J. J. WEISS
3654
prepared by anionic polymerization. The others were fractions of free-radical-initiated polymers polymerized in solution to a conversion of less than 5%.
Acknowledgment. We are grateful to the Deutsche Forschungsgemeinschaft for a research grant (to R. L. D) and for financial assistance.
An Electron Spin Resonance Study of the
y
Radiolysis and the Photolysis
of Frozen Ammonia-Water Systems
by B. S. Al-Naimy, P. N. Moorthy, and J. J. Weiss Laboratory of Radiation Chemistry, School of Chemistry, The University, Newcastle-upon-Tyne, England (Received July 11 1966)
The radical formed on y irradiation of frozen ammonia-water systems a t 77°K has been identified as the NH2 radical. The yields of NH2 radicals have been studied as a function of the ammonia concentration in these systems. Possible explanations have been offered for the dependence of the yields G(NH2) on the ammonia concentration and also for the different shapes of the esr signal of the WH2radical in the different ammonia-water matrices. The effect of electron acceptors on the NH2 radical yield in the y radiolysis of frozen ammonia-water systems has been studied. Somewhat similar observations have been made in the ultraviolet photolysis of these systems at 77°K.
Introduction Radiolysis of ice and frozen aqueous solutions has been the subject of several investigations by esr technique in this laboratory1i2 and elsewhere.3-7 It has been concluded8.9 that the OH radicals formed during irradiat'ion at 770K do not enter into reactions at this temperature and are therefore trapped in pure ice, whereas the H atoms are stable only in the presence of oxy anion solutes such as the sulfates and phosphates. The species resulting from the primary action of radiation on ice, viz., the electrons and holes (or the negative and positive polarons'O), on the other hand, are very reactive and in pure ice undergo a recombination reaction which is kinetically of the first order. Paramagnetic species formed by reaction of the electron or hole with the solutes are observed only when solutes capable of reacting with both these species are present. The investigation of the radiolysis of frozen aqueous solutions of ammonia and of frozen The Journal of Physical Chemistry
anhydrous ammonia was undertaken as an extension of the study of the radiolysis of other frozen systems.lt2 As will be evident from the later sections of this paper, (1) L. Kevan, P. N. Moorthy, and J. J. Weiss, Nature, 199, 689 (1963); J. A m . Chem. Soc., 86, 774 (1964). (2) P.N.Moorthy and J. J. Weiss, Nature, 201, 1317 (1964); Phil. M ~ . [SI , io, 659 (1964). (3) S.Siegel, L. H. Baum, S. Skolnik, and J. M. Flournoy, J . Chem. Phys., 32, 1249 (1960). (4) S. Siegel, J. M. Flournoy, and L. H. Baum, ibid., 34, 1782 (1961). (5) R.Livingston, H.Zeldes, and E. H. Taylor, Discussions Faraday SOC.,19, 166 (1965). (6) D. Schulte-Frohlinde and K. Eiben, 2. Naturforsch., A17, 445 (1962). (7) B. G. Ershov, A. K. Pykaev, P. I. Glazunov, and Y . Cpytsin, Dokl. Akad. Nauk SSSR, 149, 363 (1963). (8) P. N.Moorthy, Ph.D. Thesis, University of Durham, England, June 1965. (9) P. N. Moorthy and J. J. Weiss, Advances in Chemistry Series, No. 50,American Chemical Society, Washington, D. C., 1965,p 180. (10) J. J. Weiss, Nature, 186, 751 (1960); 199, 589 (1963).
