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Statistical theory of volume effects in macromolecules
STATISTICAL THEORY OF VOLUME EFFECTS IN MACROMOLECULES* A. K. KRON and O. B. PTITSYN Institute of high-molecular-weight compounds, U.S.S.R. Academy of Sciences
(Received 20 September
1961)
IN ORDER to calculate the thermodynamic functions and dimensions of macromolecules in solution it is necessary to know the distribution function f(rpt ) of the distance r~t between the segments numbered p and t. As a means for the approximate calculation of this function m a y be used the relation connecting it with functions of higher orders, i.e. with functions giving the distribution of probabilities between three, four, or larger numbers of segments. The distribution function f(rpt ) can be expressed in the form:
f (rpt) =Ae -t'(rpO/kT,
(1)
where F(rpt ) is the free energy of a macromolecule with a fixed distance rpt and A is a standardized constant. Fixman [1] proposed an approximate expression for the free energy F(r~t ) of a molecule with a fixed distance between the ends rl~v ---h between the segments of which forces act which rapidly fall off with the distance F(h) =
vFqh(O,j)+Fo(h),
(2)
i
where v =5[1--e-~(O/kT]-dr
(3)
is the effective excluded volume of a segment of the chain, U(r) is the potential energy of the interaction between the segments, Fo(h ) is the free energy of the macromelecule in an ideal solvent (Flory's ~ point) [2]) in which v = 0 , and qa(Oi~) is the arbitrary probability of the collision of the ith and j t h segments with fixed h. The present paper is devoted to obtaining a more accurate expression for the free energy. Let the first k--1 segments be "discharged", i.e. not interact with one another and with other segments and let the distance between the segments p and t be fixed. The corresponding free energy F(rpt; ]c--l) can be expressed b y means of a statistical integral
F(rpt; k--1)=--kT
lnf
fo{N}exp t_--/=k ~=~+~
*Vysokomol. soyed. 5: No. 3, 412-416. 1075
1076
A . K . KRON a n d O. B . PTITSYN
where/0{N} is the distribution function for all the N segments in the absence of any interaction between the segments and Ui~ is the energy of interaction of the ith and j t h segments. * The integration is carried out with respect to the coordinates of all the segments at a fixed distance between the segments p and t. Having represented the exponential factor corresponding to the interaction of the kth segment with all the other charged segments in the form of a polynomial (cf. [1]), we obtain:
Z=k+l
+
Z
exp
--
l, n = k + l
Z
_,
drpt
i=k+ l j=i + l
,
(5)
where @o= 1--exp [-- UJkT]. Since the forces ofinteractio~ between segments are of short range, ~ii ~-v6 (ro), where v is expressed by equation (3) and 6 is the delta function. The third, fourth, etc., members in the series (5) correspond to collisions of three, four, and more segments, respectively, at a single point and they are considerably smaller than the second member. Consequently, we m a y limit ourselves to only the first two members of the series:
F(rpt;k--1)------kTln
{
["
F
N
N--1N
- v F, ffo{N}a(rk,3exp|--X
F,
U ( r o ) ] ~ d{N} _,
Li=k+11=i+l
k+l
=--kTln
1--v
U(ri~)]d{N } -. kT dr~t
I7--1N
ff0{N}exp|-Z Z Li=k+li=i+l
2
q,,t(O~t; k)
kT ]~ drpt
--kTlnIL{N}
l=k+l
x exp
-- 2
2
U (rlj)/kT d {N}/drpt.
(6)
i=k+lj=i+l
Here q,~t(Okl ; k) is the arbitrary probability of the collision of the kth and /th segments in the chain where the first k segments are "discharged" and -+
segments p and t are at a distance rpt. Equation (6) can be written in the form:
F(r~;k)--F(r~t;k--1)=kTln
1--v ~ q,pt(Okt;k) .
(7)
/=k+l
Summing equ~tion (7) with respect to k from 1 to N--1, we obtain • W e a s s u m e , as usual, t h a t t h e e n e r g y o f t h e c h a i n is e q u a l t o t h e s u m o f t h e e n e r g i e s of interaction of the various pairs of segments.
Statistical theory of volume effects in macromolecules
F(rpt;N--1)--F(r~t;O)---kT N-1 ~ In I 1--v k=l
~N
q,pt(O~,;k)],
1077
(8)
l=k+l
F (rpt; 0) being equal to the free energy of the actual chain F (r~t) and 2'(rpt; N - - 1) being equal to the free energy of the completely "discharged" chain Fo(rpt) hr
= - - k T lnf0(rv~ ). The magnitude v ~
qrpt(0~l; k) is small in comparison with
l=k+l
unity (it is possible to convince oneself of this, for example, by assuming t h a t the probability q,pt(Ok~;k) is Gaussian, see below). Consequently, we finally obtain hr-1
N
F(r~t)=-2,o(rpt)+kTv ~ ~ q,pt(Okl;k)
(9)
k=l l=k+l
and, in particular N-1
2,(h)=Fo(h)+kTv Z
N
Y'. qh(Okz;k).
(10)
k = l l-=k+l
We m a y use equation (10) to obtain an approximate expression for the mean square distance between the ends of a chain for large volume effects. We m a y assume approximately [3] t h a t h 2 satisfies the equation:
d ln[h3f(h)]=O.
(11)
Using equations (1) and (10) and introducing the parameters xZ=h2/h~ and a2=h~/h~(h~=Na2--the mean square distance between the ends of the macromolecule in an ideal solvent, a being the effective length of a segment of the chain), we obtain V 3
x
d
N-1
=
N
q (o t; k) .
k=l t=k+l
Equations (10) and (12) permit the free energy and mean dimensions of a macromolecule to be calculated from the probability of the collision of segments in a partially "discharged" chain. The simplest assumption is t h a t these probabilities are equal to the probabilities of collision in a completely "discharged" chain having the same distance between the ends. This assumption was made by Katchalsky and Lifson [4] in the construction of a theory of polyeleetrolytes and also by James [5] in a theory of the volume effects in uncharged chains. Then (cf., for example,
[6]) qh(Okt; k)=qh(O~5 N ) = where m -~ l-- k.
[
3N
1312
[_2~a'm-(N--miJ
3,.n, e 2~,zc(z¢-~),
(13)
1078
A.K. KI~ONand O. B. PTITSYN
Substituting equation (13) in equation (10) and passing from summation to integration, we obtain (cf. [5])
F(h) =F0(h )-}-bT. 2~/Nz--kT~/6-~ zx,
(14)
where
Fo(h ) ----const. + 3kT x~/2, z _ _ ( 3 ~'/~ ~/~rv a'
(15)
--t~J
Equation (14) shows that the relative number of conformations of a macromolecule not forbidden by volume effects decreases exponentially as the number of segments rises (since ~/Nz ~ Nv/aa), which is confirmed by mudel calculations of the fraction of non-intersecting paths carried out on electronic computers [8]. Analogously, from equations (12) and (13) we have a-- - -a
3
z,
(16)
which differs from the Katchalsky-Lifson equation [4] only by a calculation factor close to unity. Equation (16), according to which, at a>~ 1, a ~ x / N , i.e. (h--~)1/~~ N (the molecule is drawn out i n the form of a rod), is in complete disagreement with experiment. This is due to the extreme crudity of the assumption that the probabilities of collisions in partially and completely "discharged" chains are the same. In actual fact, the probability of collisions of segments in a "charged" chain is much less than in a "discharged" chain with the same distance between the ends (cf. [7]), so that this approximation leads to a marked overestimation of the role of volume effects. As was to be expected, it gives a correct result only for the first member of the expansion in z of the distribution function for h:
f (h) = e-t'(a)/kT.~fo(h ) [1--2~/Nz-]-x/~ zx+...],
(17)
which one of us derived previously [6]. Somewhat closer to the truth is the assumption that the actual chain with volume effects replaces an equivalent Gaussian chain with segments of lenght aa (instead of a), where a depends on the total number of segments in the chain [2]. In our case of a chain with k discharged segments the value of ak will be equal to a for a chain consisting of N--/c segments. With this approximation, we shall obtain instead of equation (13)
qh(Okt; k)=\2ua2]
[al(l_k)[~(N--1)q-lc]J
e2"I:k'(s-k)+*~ E,k'(N-1)+kl (18)
Substituting equation (18)in equation (12), replacing summation by integration, and integrating up to l, we obtain
Statistical theory of volume effects in macromolecules
3
0 ak L ~ k ( 1 - - u ) + u
3a~j
2
~ / 2u[~(1--u)+u]
1079
' (19)
where (I)(x)----(2/~/~) S exp (--£) dt -- the probability integral. I f the dependence of 0
a on the number of segments in the chain is approximated b y the power function a 2 ~ N , then ~ = ~ (1--k/Ny. Using this equation, we obtain for the asymptotic ease, when ~ > 1 ~+.
=
~'/,
[~(l--s)] (l+e) sin
.
(20)
- l-Se
Equation (20)was obtained from equation (19) with the assumption that at > 1 the main contribution is given b y the first member in the square brackets and that over the whole integration interval the probability integral is close to unity. Frome equation (21) it follows that e_~ x / 5 - - - 2 ~_0.24. As is well known, Flory's theory [2] in which the real macromolecule is replaced b y a cloud of segments distributed relative to the centre of gravity b y a Gaussian law, leads to the equation as_~a =const. z, from which, at ~>> 1, ~5 ~ ~/~ i.e. e=0.20. We see that even with Flory's assumption of an effective Gaussian distribution function for chains with volume effects, consideration of the bond between the segments in the chain leads to a more marked dependence of ~ on N than a s ~ N °'2°, which is confirmed by experiment (cf. [9]). An analogous result (with e----0.33) was obtained previously by one of us by another method [9]. Replacing a chain of N - - k charged segments b y an effective Gaussiar~ chain with a segmental length of ~ka is not accurate for other reasons. In the first place, the volume effects increase mainly the mean distances between the remote segments and affect the distances between neighbouring segments comparatively little (see, for example [6]), so that the probability of collisions between neighbouring segments must i~gure more markedly in equation (13) than in equation (18), i.e. it must be greater than that calculated b y equation (18). In the second place, the volume effects make the distribution of distances about the mean values narrower (in comparison with the effective Gaussian function), which diminished the probability of collision (at a given r~ 2) as compared with equation (18). Thus, it follows from equation (17) (valid for small values of z) that f (0)=f0(0) (1--4z), while the effective Gaussian function would give f(O) =fo(O)/a a =f0(O)(1 -- 2z). Since the probability of collisions of segments decreases fairly rapidly with an increase in the mean distance between them, as can be seen from equations (13)
1080
A.K. KRONand O. B. PTITSYN
and (18), it is possible to consider t h a t the first effect, relating to neighbouring segments, will predominate over the second effect. This means t h a t in actual fact the probability of collisions of segments is greater than is given by equation (18), which may lead to an even more marked dependence of a on N than a2~N0"*4. We m a y note t h a t equations (9) and (10) are accurate and give a sound basis for obtaining subsequent approximations of the theory, takfng the non-Gaussian nature of the distribution function into account. CONCLUSIONS
(1) A general statistical method has been proposed for investigating volume effects in polymer chains in which the free energy of the macromolecules is expressed by means of the probability of the collision of segments in a partially "discharged" chain. (2) An approximate expression has been obtained for the expansion factor a of the chain based on a calculation of the probability of collisions for an effective Gaussian chain. (3) I t has been shown t h a t with this approximation a 2 ~ M °'~4 (at a ~ 1) while according to Flory's theory a s ~ M °'~°. Translated by B. J. ~ZZARD REFERENCES
1. M. FIXMAN, J. Chem. Phys. 23: 1656, 1955 2. P. FLORY, Principles of Polymer Chemistry, N.Y., 1953 3. J, HERMANS and J. OVERBEEK, Ree. tray. chim. 67: 761, 1948 4. A. KATCHALSKYand S. LIFSON, J. Polymer Sci. 11: 409, 1953 5. H. JAMES, J. Chem. Phys. 21: 1628, 1953 6. O. B. PTITSYN, Vysokomol soyed. 1: 715, 1959 7. S. LIFSON, J. Polymer Sci. 23: 431, 1957 8. F. WALL and J. ERPENBECK, J. Chem. Phys. 30: 634, 1959 9, O. B. PTITSYN, Vysokomol. soyed. 3: 1673, 1961
(Received 20 September
1961)
IN ORDER to calculate the thermodynamic functions and dimensions of macromolecules in solution it is necessary to know the distribution function f(rpt ) of the distance r~t between the segments numbered p and t. As a means for the approximate calculation of this function m a y be used the relation connecting it with functions of higher orders, i.e. with functions giving the distribution of probabilities between three, four, or larger numbers of segments. The distribution function f(rpt ) can be expressed in the form:
f (rpt) =Ae -t'(rpO/kT,
(1)
where F(rpt ) is the free energy of a macromolecule with a fixed distance rpt and A is a standardized constant. Fixman [1] proposed an approximate expression for the free energy F(r~t ) of a molecule with a fixed distance between the ends rl~v ---h between the segments of which forces act which rapidly fall off with the distance F(h) =
vFqh(O,j)+Fo(h),
(2)
i
where v =5[1--e-~(O/kT]-dr
(3)
is the effective excluded volume of a segment of the chain, U(r) is the potential energy of the interaction between the segments, Fo(h ) is the free energy of the macromelecule in an ideal solvent (Flory's ~ point) [2]) in which v = 0 , and qa(Oi~) is the arbitrary probability of the collision of the ith and j t h segments with fixed h. The present paper is devoted to obtaining a more accurate expression for the free energy. Let the first k--1 segments be "discharged", i.e. not interact with one another and with other segments and let the distance between the segments p and t be fixed. The corresponding free energy F(rpt; ]c--l) can be expressed b y means of a statistical integral
F(rpt; k--1)=--kT
lnf
fo{N}exp t_--/=k ~=~+~
*Vysokomol. soyed. 5: No. 3, 412-416. 1075
1076
A . K . KRON a n d O. B . PTITSYN
where/0{N} is the distribution function for all the N segments in the absence of any interaction between the segments and Ui~ is the energy of interaction of the ith and j t h segments. * The integration is carried out with respect to the coordinates of all the segments at a fixed distance between the segments p and t. Having represented the exponential factor corresponding to the interaction of the kth segment with all the other charged segments in the form of a polynomial (cf. [1]), we obtain:
Z=k+l
+
Z
exp
--
l, n = k + l
Z
_,
drpt
i=k+ l j=i + l
,
(5)
where @o= 1--exp [-- UJkT]. Since the forces ofinteractio~ between segments are of short range, ~ii ~-v6 (ro), where v is expressed by equation (3) and 6 is the delta function. The third, fourth, etc., members in the series (5) correspond to collisions of three, four, and more segments, respectively, at a single point and they are considerably smaller than the second member. Consequently, we m a y limit ourselves to only the first two members of the series:
F(rpt;k--1)------kTln
{
["
F
N
N--1N
- v F, ffo{N}a(rk,3exp|--X
F,
U ( r o ) ] ~ d{N} _,
Li=k+11=i+l
k+l
=--kTln
1--v
U(ri~)]d{N } -. kT dr~t
I7--1N
ff0{N}exp|-Z Z Li=k+li=i+l
2
q,,t(O~t; k)
kT ]~ drpt
--kTlnIL{N}
l=k+l
x exp
-- 2
2
U (rlj)/kT d {N}/drpt.
(6)
i=k+lj=i+l
Here q,~t(Okl ; k) is the arbitrary probability of the collision of the kth and /th segments in the chain where the first k segments are "discharged" and -+
segments p and t are at a distance rpt. Equation (6) can be written in the form:
F(r~;k)--F(r~t;k--1)=kTln
1--v ~ q,pt(Okt;k) .
(7)
/=k+l
Summing equ~tion (7) with respect to k from 1 to N--1, we obtain • W e a s s u m e , as usual, t h a t t h e e n e r g y o f t h e c h a i n is e q u a l t o t h e s u m o f t h e e n e r g i e s of interaction of the various pairs of segments.
Statistical theory of volume effects in macromolecules
F(rpt;N--1)--F(r~t;O)---kT N-1 ~ In I 1--v k=l
~N
q,pt(O~,;k)],
1077
(8)
l=k+l
F (rpt; 0) being equal to the free energy of the actual chain F (r~t) and 2'(rpt; N - - 1) being equal to the free energy of the completely "discharged" chain Fo(rpt) hr
= - - k T lnf0(rv~ ). The magnitude v ~
qrpt(0~l; k) is small in comparison with
l=k+l
unity (it is possible to convince oneself of this, for example, by assuming t h a t the probability q,pt(Ok~;k) is Gaussian, see below). Consequently, we finally obtain hr-1
N
F(r~t)=-2,o(rpt)+kTv ~ ~ q,pt(Okl;k)
(9)
k=l l=k+l
and, in particular N-1
2,(h)=Fo(h)+kTv Z
N
Y'. qh(Okz;k).
(10)
k = l l-=k+l
We m a y use equation (10) to obtain an approximate expression for the mean square distance between the ends of a chain for large volume effects. We m a y assume approximately [3] t h a t h 2 satisfies the equation:
d ln[h3f(h)]=O.
(11)
Using equations (1) and (10) and introducing the parameters xZ=h2/h~ and a2=h~/h~(h~=Na2--the mean square distance between the ends of the macromolecule in an ideal solvent, a being the effective length of a segment of the chain), we obtain V 3
x
d
N-1
=
N
q (o t; k) .
k=l t=k+l
Equations (10) and (12) permit the free energy and mean dimensions of a macromolecule to be calculated from the probability of the collision of segments in a partially "discharged" chain. The simplest assumption is t h a t these probabilities are equal to the probabilities of collision in a completely "discharged" chain having the same distance between the ends. This assumption was made by Katchalsky and Lifson [4] in the construction of a theory of polyeleetrolytes and also by James [5] in a theory of the volume effects in uncharged chains. Then (cf., for example,
[6]) qh(Okt; k)=qh(O~5 N ) = where m -~ l-- k.
[
3N
1312
[_2~a'm-(N--miJ
3,.n, e 2~,zc(z¢-~),
(13)
1078
A.K. KI~ONand O. B. PTITSYN
Substituting equation (13) in equation (10) and passing from summation to integration, we obtain (cf. [5])
F(h) =F0(h )-}-bT. 2~/Nz--kT~/6-~ zx,
(14)
where
Fo(h ) ----const. + 3kT x~/2, z _ _ ( 3 ~'/~ ~/~rv a'
(15)
--t~J
Equation (14) shows that the relative number of conformations of a macromolecule not forbidden by volume effects decreases exponentially as the number of segments rises (since ~/Nz ~ Nv/aa), which is confirmed by mudel calculations of the fraction of non-intersecting paths carried out on electronic computers [8]. Analogously, from equations (12) and (13) we have a-- - -a
3
z,
(16)
which differs from the Katchalsky-Lifson equation [4] only by a calculation factor close to unity. Equation (16), according to which, at a>~ 1, a ~ x / N , i.e. (h--~)1/~~ N (the molecule is drawn out i n the form of a rod), is in complete disagreement with experiment. This is due to the extreme crudity of the assumption that the probabilities of collisions in partially and completely "discharged" chains are the same. In actual fact, the probability of collisions of segments in a "charged" chain is much less than in a "discharged" chain with the same distance between the ends (cf. [7]), so that this approximation leads to a marked overestimation of the role of volume effects. As was to be expected, it gives a correct result only for the first member of the expansion in z of the distribution function for h:
f (h) = e-t'(a)/kT.~fo(h ) [1--2~/Nz-]-x/~ zx+...],
(17)
which one of us derived previously [6]. Somewhat closer to the truth is the assumption that the actual chain with volume effects replaces an equivalent Gaussian chain with segments of lenght aa (instead of a), where a depends on the total number of segments in the chain [2]. In our case of a chain with k discharged segments the value of ak will be equal to a for a chain consisting of N--/c segments. With this approximation, we shall obtain instead of equation (13)
qh(Okt; k)=\2ua2]
[al(l_k)[~(N--1)q-lc]J
e2"I:k'(s-k)+*~ E,k'(N-1)+kl (18)
Substituting equation (18)in equation (12), replacing summation by integration, and integrating up to l, we obtain
Statistical theory of volume effects in macromolecules
3
0 ak L ~ k ( 1 - - u ) + u
3a~j
2
~ / 2u[~(1--u)+u]
1079
' (19)
where (I)(x)----(2/~/~) S exp (--£) dt -- the probability integral. I f the dependence of 0
a on the number of segments in the chain is approximated b y the power function a 2 ~ N , then ~ = ~ (1--k/Ny. Using this equation, we obtain for the asymptotic ease, when ~ > 1 ~+.
=
~'/,
[~(l--s)] (l+e) sin
.
(20)
- l-Se
Equation (20)was obtained from equation (19) with the assumption that at > 1 the main contribution is given b y the first member in the square brackets and that over the whole integration interval the probability integral is close to unity. Frome equation (21) it follows that e_~ x / 5 - - - 2 ~_0.24. As is well known, Flory's theory [2] in which the real macromolecule is replaced b y a cloud of segments distributed relative to the centre of gravity b y a Gaussian law, leads to the equation as_~a =const. z, from which, at ~>> 1, ~5 ~ ~/~ i.e. e=0.20. We see that even with Flory's assumption of an effective Gaussian distribution function for chains with volume effects, consideration of the bond between the segments in the chain leads to a more marked dependence of ~ on N than a s ~ N °'2°, which is confirmed by experiment (cf. [9]). An analogous result (with e----0.33) was obtained previously by one of us by another method [9]. Replacing a chain of N - - k charged segments b y an effective Gaussiar~ chain with a segmental length of ~ka is not accurate for other reasons. In the first place, the volume effects increase mainly the mean distances between the remote segments and affect the distances between neighbouring segments comparatively little (see, for example [6]), so that the probability of collisions between neighbouring segments must i~gure more markedly in equation (13) than in equation (18), i.e. it must be greater than that calculated b y equation (18). In the second place, the volume effects make the distribution of distances about the mean values narrower (in comparison with the effective Gaussian function), which diminished the probability of collision (at a given r~ 2) as compared with equation (18). Thus, it follows from equation (17) (valid for small values of z) that f (0)=f0(0) (1--4z), while the effective Gaussian function would give f(O) =fo(O)/a a =f0(O)(1 -- 2z). Since the probability of collisions of segments decreases fairly rapidly with an increase in the mean distance between them, as can be seen from equations (13)
1080
A.K. KRONand O. B. PTITSYN
and (18), it is possible to consider t h a t the first effect, relating to neighbouring segments, will predominate over the second effect. This means t h a t in actual fact the probability of collisions of segments is greater than is given by equation (18), which may lead to an even more marked dependence of a on N than a2~N0"*4. We m a y note t h a t equations (9) and (10) are accurate and give a sound basis for obtaining subsequent approximations of the theory, takfng the non-Gaussian nature of the distribution function into account. CONCLUSIONS
(1) A general statistical method has been proposed for investigating volume effects in polymer chains in which the free energy of the macromolecules is expressed by means of the probability of the collision of segments in a partially "discharged" chain. (2) An approximate expression has been obtained for the expansion factor a of the chain based on a calculation of the probability of collisions for an effective Gaussian chain. (3) I t has been shown t h a t with this approximation a 2 ~ M °'~4 (at a ~ 1) while according to Flory's theory a s ~ M °'~°. Translated by B. J. ~ZZARD REFERENCES
1. M. FIXMAN, J. Chem. Phys. 23: 1656, 1955 2. P. FLORY, Principles of Polymer Chemistry, N.Y., 1953 3. J, HERMANS and J. OVERBEEK, Ree. tray. chim. 67: 761, 1948 4. A. KATCHALSKYand S. LIFSON, J. Polymer Sci. 11: 409, 1953 5. H. JAMES, J. Chem. Phys. 21: 1628, 1953 6. O. B. PTITSYN, Vysokomol soyed. 1: 715, 1959 7. S. LIFSON, J. Polymer Sci. 23: 431, 1957 8. F. WALL and J. ERPENBECK, J. Chem. Phys. 30: 634, 1959 9, O. B. PTITSYN, Vysokomol. soyed. 3: 1673, 1961