REVISEDTHEORY OF HIGH P O L Y M E R
Sept. 5 , 1964 [ COKTRIBUTION
FROM THE
KODAKRESEARCH LABORATORIES, ROCHESTER, SEW
MENLOP A R K ,
3535
SOLUTIONS
YORK, AND STANFORD RESEARCH INSTITUTE,
CALIFORXIA]
A Revised Theory of High Polymer Solutions' BY MAURICEL. HUGGINS RECEIVED APRIL 16, 1964 Theoretical equations, in closed form and as expansions in powers of the concentration, have been deduced for the enthalpy and entropy parts of the interaction coefficient for solutions of linear high polymers. These equations relate experimentally observable quantities t o molecular and intermolecular properties. They furnish a reasonable basis for the approximately opposite concentration dependence of the enthalpy and entropy functions. I t is expected t h a t these results will prove useful in connection with various properties of polymer solutions and gels a t intermediate and high concentrations
Introduction This paper reports an extension of the theory of the thermodynamic properties of solutions of long-chain polymers, previously deduced by the a u t h ~ r . ~ I-t~s chief aims are to relate experimentally measurable quantities as closely as possible t o molecular structures and intermolecular interactions and to obtain equations which will give more quantitative agreement with experiment than do previous theoretical equations, especially a t moderate and high concentrations of polymer. T h e present paper deals exclusively with the theory. Application of the results to experimental d a t a is in progress and will be reported later. Following previous theoretical developments by the Flory, and others, the partial molal Gibbs free energy of mixing of the solvent, in a solution of linear polymer of a single type in a single solvent, is given by the equation
I t is known t h a t x is a function of both the enthalpy and the entropy of the solution. I t is convenient for our purpose to write X =
( 1 ) E x c e p t f o r minor changes a n d a d d i t i o n s , t h i s t h e o r y was developed while t h e a u t h o r was on t h e staff of t h e R e s e a r c h Laboratories of t h e E a s t m a n K o d a k C o , R o c h e s t e r , K. Y H e is now o n t h e s t a 5 of Stanford R e search I n s t i t u t e . Menlo P a r k , Calii A preliminary r e p o r t of t h i s work was presented a t t h e s y m p o s i u m honoring Prof Joel H . Hildehrand o n t h e occasion of his 8 0 t h b i r t h d a y , Berkeley, Calif , S e p t 1 2 , 1961 ( 2 ) . M I . Huggins, J P h y s C h r m . , 46, 151 (1942). ( 3 ) M L . Huggins. A n n .V. Y Acari S c i , 41, 1 (1942). ( 4 ) M I . Huggins, J . A m Chem S o c , 64, 1712 (1942). ( 5 ) M 1- Huggins. J P h y s Coiiord Chem , 62, 248 ( 1 9 4 8 ) . (6) %I I. Huggins. J P o i y m r r .Sci , 16, 209 (195,5), ( 7 ) M 1, Huggins, "Physical C h e m i s t r y of H i g h Polymers." J o h n Wiley a n d S o n s , Inc , New Y o r k , N . Y , 19.i8, C h a p t e r 6 18) P J F l o w J C h r m P h y s , 10, 5 1 (1942). (9) P J Flory. : b d , l a , 425 11944) ( 1 0 ) P. J P l o r y . "Principals oi Polymer C h e m i s t r y . " Cornell University Press, I t h a c a , S Y . , -1953
+
(2)
xs
and to consider the enthalpy contribution, entropy contribution, xs,separately.
Xh,
and the
Theoretical Development : Enthalpy and Energy Rather than dealing with the enthalpy of mixing directly, i t is simpler to deal first with the energy of mixing, assuming zero volume change on mixing, and then, following Hildebrand and Scott, l 1 to estimate the corrections for volume change by means of the relationships AFi AI?l
=
=
AAi
AEl(1
+ aBT)
(3)
(4)
Here, A A 1 is the Helmholtz free energy of mixing and CY, is the coefficient of cubical expansion of the solution, approximated by 0.8
Here, and v2 are the partial molal volumes of solvent and solute, respectively, & and #q are their volume fractions in the solution, and x is a parameter, which depends on the solute and solvent and the temperature, b u t not appreciably on the molecular weight of the polymer, as long as i t is high. I t does depend on the concentration, but approaches constancy as the approaches zero. Various exconcentration (i.e., perimentallv measurable quantities of interest can be related to AFI and so to x. Considering x as concentration dependent is of course equivalent to expanding x"&, eq. 1 into a series, with additional terms etc. The problem to be solved is the quantitative evaluation of x, including its concentration dependence, in terms of molecular and intermolecular properties.
Xh
=
CY141
+
CY242
(5)
We are concerned with the energy of molecular interactions in a solution of chain molecules, and with the dependence of that energy on the molecular properties and on the concentration. Except when Coulombic forces are involved, it seems reasonable to consider the molecular interaction energies as depending solely on the "contacts" between adjacent molecules. T o put this concept on a quantitative basis, we consider that each molecule has a molecular surface of definite b u t unspecified area and that the interaction energy associated with contact between two adjacent molecules depends on the kinds of molecules and on the area'of mutual contact. I t is of course necessary to deal with averages and to make allowance for intramolecular contacts and for the portions of the surface areas of molecules which are not in contact with surface areas of the same or other molecules, and to allow for departures from perfect randomness of mixing, resulting from differences in the interaction energies between different types of pairs of molecules. We designate by u1 and u2 the average "molecular surface areas" of solvent and solute (polymer) niolecules, respectively. By u l l , u12, and ulo we designate the average surface areas per solvent molecule which are "in contact" with other solvent molecules, solute molecules, and nothing, respectively. By u21, u l z , i n t u22,ext, and u20 we denote the average surface areas of a (11) J H Hildehrand a n d K I. S c o t t , "Solubility o f S o n ? l r c t r ~ i l y f e s . " 3rd E d , Reinhold Puhlishing C o r p , S e w Y o r k , S Y , I!l.iO
MAURXCE L. HUGGINS
3536
solute molecule in contact with solvent molecules, other portions of the surface of the same solute polymer molecule, other solute molecules, and nothing, respectively. Then
+ + + + +
u1 =
u11
uz = ~ 2 1
LT22.int
(6)
u10
012
u22,ext
uzo
=
- u10 = Q l l Z O- U 2 2 , i n t =
Ul
+
u12
+
Here, m is an empirical constant and k S h is a constant for a given system a t a given temperature, approximately unity for low polymers and decreasing as the average molecular weight increases. The volume fraction of solvent in the solution is related to the molecular volumes (211, v2) by the equation
(7)
We assume U I , U Z , U I O , and uzO to be ind,ependent of concentration. Also, although i t is likely t h a t uZ2,int varies with concentration in some instances, we neglect such variation and take i t also as constant. We can then define the “effective surface areas,” ~ 1 and ’ u2’, by the equations Ul’
Vol. 86
(8)
~ 2 ’= ~2 - ~ ~ 2 1 u22,ext (9) If N1 and N2 are the numbers of solvent and solute molecules in the solution, respectively N l U l 2 = N2LT2, (10) Leaving out of consideration all the internal energies of the molecules which should be independent of concentration, the energy of the solution is the sum of the energies of intermolecular contact. We designate by €11, €22, and t z l the average interactioii energies per unit area of contact, for the three types of interinolecular contact. These should each be independent of concentration at a given temperature. (For intermolecular attraction, these are negative.) Let p 2 1 and p ~ be ~the probabilities , ~ ~ t h~ a t a small area of the surface of a polymer molecule, which is in contact with another molecule, is in contact with a solvent and with another solute molecule, respectively. If the two types of molecule were similar in size, shape, rigidity, and intermolecular contact energy (ix.,€21 = we should expect these probabilities to be proportional to the available molecular surface areas of the two types. Hence
If, however, the solute molecules are flexible linear polymer molecules, the presence of one contact between such molecules increases the probability of two or more contacts. To allow for this we may insert a “multiple contact factor,” k,, in the numerator on the right side of eq. 11. (This factor corresponds to (1 f)- - I in ref. 7 For a fuller discussion, see ref. 6, pp. 216217.) For present purposes, we assume k,, to be independent of concentration. For high polymer solutions, another factor, f s h , should be included in the numerator of eq. 11 to allow for the fact t h a t the interior segments of a convoluted molecule are partially shielded from contact with interior segments of other polymer molecules. This may be very important in dilute solutions, but the shielding should be negligible in concentrated solutions when the polymer molecules interpenetrate each other to a large extent. Tentatively, we shall assume the shielding factor to depend on concentration according to the equation
At infinite dilution, f s h equals k s h ; in pure polymer, f s h is unity. If the attraction, per unit surface area, between like molecules is greater than that between unlike molecules, the ratio of eq. 11 should be increased. I t seems reasonable to approximate this increase by insertion of a factor W
=
e x p (k,Ae kT) = 1
+ k,Ae +
. . .
(14)
where A €
= 2621 -
€11
-
€22
(15)
kg is Boltzmann’s constant, T is the absolute temperature, and k , is a factor t h a t depends on chain flexibility and other structural features. We thus have, for polymer solutions
From this and eq. 9, one readily obtains
The partial molal energy of the solvent is related to the energy of mixing by the equation
where -V*is Avogadro’s number. Performing the indicated differentiation yields the result
REVISEDTHEORY OF HIGH POLYMER
Sept. 5 , 1964
3537
SOLUTIONS
The ratio of molecular volumes, vl/vZ, can reasonably be taken equal to the ratio of partial molal volumes and to the ratio of contacted molecular surface areas
3 -- P2 Pz
vz
u1
UZf
+
I
(22) ~22.1nt
Then
AB1
=
(34)
NAP1uzkextkmcWAE422[fsh -k m(1 - f s d 4 z l 2 P ~ [l (1 - kextkmcfshW)4212
and
(23) ash
or /1
-
f .\
1
\
and uz ~
u2 - u z o
(26)
which equals the average fraction of contacted polymer molecule surface which is in contact with portions of other molecules. Expansion of eq. 24 in powers of $2 gives the series
9 (imz - i m
- 1)
(35)
Theoretical Development : Entropy T h e entropy with which we are concerned is related t o the randomness of placing of the molecules and molecular segments in the solution. It is convenient to consider each polymer molecule as composed of n, like rigid segments. We imagine these molecules as being placed in the volume, V , t o be occupied b y the solution, one segment a t a time for t h e first molecule, then one segment a t a time for the second molecule, and so on until all the N Z polymer molecules have been placed. The N1 solvent molecules (presumed to be rigid) are then hypothetically inserted one a t a time. The entropy ( S ) of the solution is related t o t h e randomness of placing of the polymer segments and solvent molecules by the equation
Here, v , designates ~ the randomness of placing the first rigid segment of the i t h solute molecule, vi) denotes t h e randomness of placing the j t h rigid segment (with j > 1) of the zth polymer molecule, and v 1 represents the randomness of placing the Ith solvent molecule. The randomness of placing the first segments can be expressed b y the relationship
+ 3 - 6mkeXtkm,W
6)kextkmcWksh
(&
The function x h o is the same as Flory’s “heat parameter,” ~1 (see ref. 10, p. 5 2 2 ) .
with
kext =
= m
+
vil = - 1
z
- k2vz-
Vr
(27)
v
f
higher terms
vz2i2 v23i3 VZ’ 7 1 3 ’ et..)] ~~
From eq. 4 and 24, the enthalpy contribution to the interaction coefficient x is deduced t o be
(37)
with vr representing the volume of the segment, equal to v z / n , if all segments are alike, and with kz a constant, of the order of magnitude of a small integer, which represents the average ratio of the volume excluded from occupancy by the center of the segment being placed, as a result of the presence of an already placed polynier molecule, to the volume actually occupied by t h a t placed molecule. Equation 37 can also be written Yil
= lvznr[ - 1-
62
(‘2); +
terms involving
Vol. 86
MAURICEL. HUGGINS
3538
ratio U Z Z , ~ ~ ~ / ( T ~ ’ ,computed from eq. 9 and 17 with substitution of i for iVz. Thus vtj u22,ext
Pi
=
=
v t o ( l - kspt)
-
U2l
u21
(7 = () ; ); = uzl
-
(45)
(2;)
i
For large Nz (46)
xz
r l n ( 1 -
f
k242.
)
. . . di
(40)
where k, is a proportionality constant and
I t can be shown t h a t for flexible high polymers the higher terms in the expansion are negligible. Then
uz
K p = ykmcfahW
(47)
u1
Hence
-
_ _kzNz _ 42 2
* terms involving 42, etc. (42)
For the solvent molecules, to a sufficient degree of approximation =
(n, - 1)Nz In
+ (n, -1)
VO
([Nz +
Hence NI
In
rl[
v i =In
NI!
=
N~ In
~1
-
~1
(44)
For large N Zand N1
i = l
The randomness of location of each rigid polymer segment, other than the first, is just its randomness of orientation relative to the preceding segment. This randomness is, of course, reduced by blocking by solvent molecules, other segments of the same polymer molecule, and other polymer molecules. Let us define v0 as the average randomness of orientation of a rigid segment of a solute molecule in infinitely dilute solution, ie., in contact only with solvent molecules and other portions of the same polymer molecule. ( I t will be recalled that we have assumed t h a t the chance of contact with another portion of the same molecule does not change with the concentration). The averaging is over all values of j , as well as over all configurations of the polymer molecule, all distributions and orientations of neighboring solvent molecules, etc. At finite concentrations the average randomness of orientation is changed (usually reduced) from the value U , by the presence in the vicinity of ( i e . , blocking by) other polymer molecules. I t seems reasonable to assume that the blocking effect (averaged for any segment of the zth molecule) is proportional to the fraction of noninternally contacted surface which is in contact with other polymer molecules, that is, to the
N z In N z - N2
(49)
In N l ! = N 1 In Nl - NI
(50)
In Nz!
=
Substitution of eq. 39,41, 44, 48,49, and 50 into eq. 36 gives
S_ -
N z In n, - N 2 In +2 -
kB
(nr - 1)Nz In
YO
+
NA1 - k z h ) In (1 - k z h ) kzh
(fir
- 1) [
~
+
+ 2
The partial molal entropy of mixing of the solvent is given by the equation
Sept. 5, 1964
REVISED THEORY OF HIGHPOLYMER SOLUTIONS
where R is the molal gas constant. relation
Making use of t h e
3539
with
we deduce -A =S
-
R
YlIn
(1 - k242)
kz Yz
+
and
The Interaction Coefficient and Its Temperature Dependence It has been customary to assume t h a t one can separate the interaction coefficient into a temperature independent part and a temperature dependent part, according to the equation
Expansion in powers of 42 leads to
x=a,+-
One would expect kz to be of the same order of magnitude as KW. Therefore, for large n,, the terms involving k2 can be neglected. Also, n, - 1 can be replaced by n,. Hence we put
Referring back t o eq. 1 and 2, we can relate AS1 to by means of the equation
- R [ln
AS1 =
41
+ (1
-
Z)~Z
4- ~
xs
~ (57) 4 ~
bx T
and then to identify these parts with the entropy and enthalpy contributions, xs and Xh (see eq. 2 ) . From the theory presented here, however, &, and hence x s , should depend on temperature, since the factor W , which enters into the expressions for $so, &’, etc., is temperature dependent (see eq. 14). Of course, if the product k,Ae is sufficiently small relative to k g l ’ , the temperature dependence of xs may be negligible. Nevertheless, i t would seem worthwhile to transfer the temperature dependent part of x s to the other component part, so obtaining theoretical expressions for the temperature independent and temperature dependent components, which can be readily measured experimentally. This has been done, neglecting all terms in powers of 1 / Z higher than the first, with the following results. ~ 1
It is now convenient to define an “entropy parameter,” $s,
by means of the relationship
$ = -1
AS1 -Pi ~ -In 41 -
( (:+$+$+ . . .) -
4zZ( R =
V$’)
=
-
+Z
42
)-
Xa
(58)
Xa
This parameter is equivalent to Flory’s “entropy parameter,” $1 (see ref. 10, p. 522), if all terms except the first, in the series expansion in eq. 58, are negligible. From eq. 56 and 58, we obtain
In [I
{-
+ (1 - k8)k,,tkmcf9~W~n/d~I (1
1
+
- k,) In (1 kcxtkmcfahW@Z/dl) (1 - ks)kextkmcfshW42/dJ1
bx (59)
Expanding into a series in powers of the volume fraction of polymer $8
=
$e”
+
$9’42
+ $8”4z2 +
.
,
.
(60)
=
b,”
+ bX’4z + bX’’6z2+ . . .
(70)
G. C. BERRYAND T. G F o x
3540
Vol. 86
can use these ratios and eq. 68, 69, 72, and 73 (putting = 0) to deduce the remaining four unknowns in these equations: K , a s h , k,, and m. Froin eq. 35, one can obtain k s h . Equation 12 will then give S s h as a function of the concentration. Equation 67 will yield n, and eq. 71 will give aL (assuming that cys has been estimated by means of eq. 5). From K and k s h , one can (eq. 33) determine the product kextkmc, b u t not the individual values of these two factors. Likewise, from eq. 34, one can obtain the product uz'Ae, b u t not uZ1and A € separately. In this way, one can determine, from experimental data, all the constants needed for substitution into the The p in eq. 71-73 is defined by the equation closed-form equations for ARI and ASl (eq. 28 and 54). If 0 is not negligible, the more complicated equations, p = - nrksk, (74) containing this quantity, must be used; another exuz I perimental quantity, such as $,,"'/$ao or bX"l/bX0, is I t is possible that some of the quantities assumed needed, unless one or more of the "unknowns" can be in the foregoing derivations to be independent of temdetermined or estimated in another way. perature do in fact vary with the temperature. If so, Testing of the equations presented in this paper, usappropriate adjustments must be made in these relaing published experimental data, has been begun. T h e tionships. results will be reported in due course. If the theory The functions deduced for X h and $s show a very and its equations should prove satisfactory, it will be similar dependence on concentration. (Compare eq. possible, from a relatively small amount of experimental 28 with eq. 59, also 31 with 62 and 32 with 63). Theredata, to deduce curves for the variation of the thermofore Xh and xs should vary oppositely, a behavior that dynamic properties of polymer solutions up to quite has often been noted empirically. In certain systems high concentrations. (At very high concentrations the relationship is approximately a rectilinear 0ne.7312,13 certain other factors6 not considered in the present The accuracy of the rectilinearity probably depends development may become important.) It is hoped, on the magnitudes of certain of the parameters. moreover, that the theory will lead to the determination of the various molecular constants which affect the Correlation with Experiment thermodynamic solution properties and so eventually From precise experimental data one can determine x and its dependence on concentration and tempera- to a better understanding of the phenomena and an ability to predict the properties of new systems. ture, hence values of $ao, h,O, and the ratios $ a ' / $ O , An extension of this theory to solutions of graft and J / a " / J / a o , b,'/b,O, and b,"/b:. If p is negligible, one block copolymers has already been published.I4 3!,
(12) H. T a k e n a k a , J. Polymer Sci., 24, 321 (1957). (13) G . Rehage a n d H. M e y s , ibid., SO, 271 (1958).
(14) M. L. Huggins. $bid., C1, 445 (1963).
[CONTRIBUTIOK FROM
THE
MELLONINSTITUTE, PITTSBURGH 13, P E N N S Y L V A N I A ]
Viscometric Tests of Excluded Volume Theories BY G. C. BERRYA N D T. G F o x RECEIVEDAPRIL15, 1964 Some aspects of viscometric tests for excluded volume theories are discussed. Attention is confined t o those systems for which the intrinsic viscosity and the molecular weight are known, possibly as a function of temperature. I t is concluded that definitive evaluation of the various theories requires d a t a a t higher molecular weights than normally studied or over a large temperature span.
Introduction I t is our purpose here to comment briefly on some aspects of viscometric tests for the various theories of the excluded volume effect in polymer coils. We will restrict our attention to studies for which only [77] and M are known, possibly as a function of temperature. Viscometric studies have recently received renewed interest because of new theoretical developments in both the hydrodynamic and thermodynamic aspects of the problem. I t is an over-simplification to consider these aspects separately, b u t the theoretical developments which attempt to include both effects simultaneously are still a t an early stage. l P 3 M K u r a t a a n d H Y a m a k a w a , J . C h e m . P h y s . , 29, 311 (1958). ( 2 ) 0 B. P t i t s y n and I. E . Eizner, Z h . F i s . K h i m . . 92, 2464 (1958).
(1)
Thermodynamic Effects on the Intrinsic Viscosity The molecular description of the hydrodynamic flow has been understood in terms of equivalent models advanced by Debye and B ~ e c h e ,Kirkwood ~ and R i ~ e m a n and ,~ and this description has been utilized to form the basis of approximations to include thermodynamic (excluded volume) effects.1 , 2 , 8 , 9 A principle result of the hydrodynamic calculations (3) See, for example, t h e review: A. Peterlin, M a k r o m o l . C h e m . , 94, 89 (1959). (4) P. Debye a n d A. M . Bueche, J . Chem. P h y r . . 16, 565 (1948) (5) J. G. Kirkwood and J. Riseman, ibid., 18, 565 (1948). (6) H. C. B r i n k m a n , P h y s i c a , 19, 447 (1947). (7) B. H. Zimm, J . C h e m . P h y s . , 1 4 , 269 (1956). ( 8 ) T. G Fox a n d P. J. Flory, J A m , C h e m . Soc., 'IS, 1904 (1951). (9) M . K u r a t a and W. H Stockmayer, A d v a n Polymcr Sci.. 9 , 196 (1963).
