Thermodynamics of conformation and deformation of linear polymeric chains in solution
Thermodynamics of conformation and deformation of linear polymeric chains in a solution is built based on the statistics of self-avoiding walks. The entropy and free energy of conformation of a polymeric chain is presented as a sum of two terms. The first one takes into account the contribution...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Thermodynamics of conformation and deformation of linear polymeric chains in solution / Yu.G. Medvedevskikh // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 219-233. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1204312017-06-13T03:04:42Z Thermodynamics of conformation and deformation of linear polymeric chains in solution Medvedevskikh, Yu.G. Thermodynamics of conformation and deformation of linear polymeric chains in a solution is built based on the statistics of self-avoiding walks. The entropy and free energy of conformation of a polymeric chain is presented as a sum of two terms. The first one takes into account the contribution of random walk and the second one takes into account the contribution of two limitations, which covered random walk and create the effect of self-organization of a polymeric chain. Deformation of the polymeric chain is considered as an equilibrious transition of Flory ball into conformational ellipsoid. The expressions for thermodynamic and elastic properties of the polymeric chain as functions of the degree of its deformation are suggested. Volumetric module, Young’s module and module of polymeric chain shift are expressed through the pressure of conformation; Poisson’s ratio depends only upon the dimension of Euclidean space. Forces and work of deformation are determined; the method of calculating the main tensions is suggested. Термодинаміка конформації і деформації лінійних полімерних ланцюгів у розчині побудована на основі статистики блукань без перехрещення. Остання виражає ентропію та вільну енергію конформації полімерного ланцюга у вигляді суми доданків, один з яких враховує внесок випадкових блукань, інший – двох обмежень, що накладаються на випадкові блукання і створюють ефект самоорганізації полімерного ланцюга. Його деформація розглянута у вигляді рівноважного переходу клубка Флорі в конформаційний елiпсоїд. Запропоновані вирази термодинамічних та пружних властивостей по- лімерного ланцюга як функцій ступеня його деформації. Об’ємний модуль, модулі Юнга та зсуву полімерного ланцюга виражаються через тиск конформації, коефіцієнт Пуассона залежить тільки від розмірності евклідового простору. Визначені сили та робота деформації, запропонований метод розрахунку головних напружень. 2001 Article Thermodynamics of conformation and deformation of linear polymeric chains in solution / Yu.G. Medvedevskikh // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 219-233. — Бібліогр.: 9 назв. — англ. 1607-324X PACS: 05.70.Ce DOI:10.5488/CMP.4.2.219 http://dspace.nbuv.gov.ua/handle/123456789/120431 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Thermodynamics of conformation and deformation of linear polymeric
chains in a solution is built based on the statistics of self-avoiding walks.
The entropy and free energy of conformation of a polymeric chain is presented
as a sum of two terms. The first one takes into account the contribution
of random walk and the second one takes into account the contribution
of two limitations, which covered random walk and create the effect of
self-organization of a polymeric chain. Deformation of the polymeric chain
is considered as an equilibrious transition of Flory ball into conformational
ellipsoid. The expressions for thermodynamic and elastic properties of
the polymeric chain as functions of the degree of its deformation are suggested.
Volumetric module, Young’s module and module of polymeric chain
shift are expressed through the pressure of conformation; Poisson’s ratio
depends only upon the dimension of Euclidean space. Forces and work of
deformation are determined; the method of calculating the main tensions
is suggested. |
| format |
Article |
| author |
Medvedevskikh, Yu.G. |
| spellingShingle |
Medvedevskikh, Yu.G. Thermodynamics of conformation and deformation of linear polymeric chains in solution Condensed Matter Physics |
| author_facet |
Medvedevskikh, Yu.G. |
| author_sort |
Medvedevskikh, Yu.G. |
| title |
Thermodynamics of conformation and deformation of linear polymeric chains in solution |
| title_short |
Thermodynamics of conformation and deformation of linear polymeric chains in solution |
| title_full |
Thermodynamics of conformation and deformation of linear polymeric chains in solution |
| title_fullStr |
Thermodynamics of conformation and deformation of linear polymeric chains in solution |
| title_full_unstemmed |
Thermodynamics of conformation and deformation of linear polymeric chains in solution |
| title_sort |
thermodynamics of conformation and deformation of linear polymeric chains in solution |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120431 |
| citation_txt |
Thermodynamics of conformation and
deformation of linear polymeric chains
in solution
/ Yu.G. Medvedevskikh // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 219-233. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT medvedevskikhyug thermodynamicsofconformationanddeformationoflinearpolymericchainsinsolution |
| first_indexed |
2025-07-08T17:52:19Z |
| last_indexed |
2025-07-08T17:52:19Z |
| _version_ |
1837102155714002944 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 219–233
Thermodynamics of conformation and
deformation of linear polymeric chains
in solution
Yu.G.Medvedevskikh
The Department of the L.V. Pisarzhevsky Institute of Physical Chemistry
of the National Academy of Sciences of Ukraine,
3a Naukova Str., 79053 Lviv, Ukraine
Received December 20, 2000
Thermodynamics of conformation and deformation of linear polymeric
chains in a solution is built based on the statistics of self-avoiding walks.
The entropy and free energy of conformation of a polymeric chain is pre-
sented as a sum of two terms. The first one takes into account the contri-
bution of random walk and the second one takes into account the contribu-
tion of two limitations, which covered random walk and create the effect of
self-organization of a polymeric chain. Deformation of the polymeric chain
is considered as an equilibrious transition of Flory ball into conformation-
al ellipsoid. The expressions for thermodynamic and elastic properties of
the polymeric chain as functions of the degree of its deformation are sug-
gested. Volumetric module, Young’s module and module of polymeric chain
shift are expressed through the pressure of conformation; Poisson’s ratio
depends only upon the dimension of Euclidean space. Forces and work of
deformation are determined; the method of calculating the main tensions
is suggested.
Key words: conformation, thermodynamics, polymeric chains,
deformation modules of elasticity, forces, work
PACS: 05.70.Ce
1. Introduction
In many aspects conformation of polymeric chains determines their dynamic and
thermodynamic properties in solution. However thermodynamics of deformation of
polymeric chains developed in works [1–7] mainly directed towards the analysis and
explanation of the elastic properties of caoutchouc has two substantial drawbacks
that make it incomplete and not rigorous. Firstly, since it is based on the statistics
of random walk (RW) of the end of the of Kuhn chain [4], the effect of exclusive
volume, which requires statistics of self-avoiding walk SAW [8] is not considered.
Secondly, using an obviously correct statement that the exterior hydrostatic pressure
c© Yu.G.Medvedevskikh 219
Yu.G.Medvedevskikh
cannot considerably influence the conformational condition of a polymeric chain in
a solution, this thermodynamics did not take into account the main parameter in
the thermodynamic relations, i.e., the volume of the conformation.
Statistics of SAW [8] describes the conformation of an ideal polymeric chain in
an ideal solution. According to the statistics, SAW limitation for self-avoidance of
trajectories of RW causes such self-organization of polymeric chain when its confor-
mation has got the structure of spherical or, in a more general case, ellipsoid shell,
that is Flory radius thick and it is diffusively smeared outside as well as inside the
polymeric blob, the core of which is statistically empty for half of Flory radius.
Conformation of ellipsoid can be considered as a deformed condition of Flory
ball that permits statistics of SAW to describe thermodynamics of conformation
and deformation of a polymeric chain in a solution from general and rigorous point
of view.
2. Entropy and free energy of conformation
Statistics of SAW on d-dimensional lattice is determined by probability w(s)
that, by its last step, the trajectory of SAW will end in one of 2d equiprobable cells
Mp(s) which have only different signs of a component vector s = (si), i = 1, d,
expression [8]
w(s) = β exp
{
−N2
/
dd/2
∏
i
|si| −
1
2
∑
i
s2i
ni
}
, (1)
in which β−1 = e (2π)d/2 is the numeral constant, N is the number of sections or
length of chain, d is the dimension of Euclidean space, n i = n+
i +n
−
i is the number of
steps in i−m direction of space including positive n+
i and negative n−
i , si = n+
i −n
−
i
is the number of effective steps.
Numbers ni, i = 1, d of steps in all d directions of space and length of chain N
are connected by the following relation
∑
i
ni = N. (2)
Probability w(s) is numerically equal to the shares of those trajectories of SAW
from common number (2d)N of trajectories RW that are possible on d-dimensional
lattice, which realize the state Mp(s). The number of such trajectories determines
thermodynamic probability W (s) of realization of the state Mp(s):
W (s) = (2d)Nw(s). (3)
We introduce a variable of metric shift of the end of chain xi, i = 1, d as semi-axis
of conformational ellipsoid to whose surface belong the states Mp(s) [8]:
xi = a|si|d
1/2. (4)
220
Thermodynamics of polymeric chains in solution
Here a is a statistical length of a section of chain.
Let us denote by σi the standard deviations of Guassian part of distribution (1):
σ2
i = a2nid. (5)
According to (2) the value of σi; is represented by the following relation
∑
i
σ2
i = a2Nd. (6)
Since s2i /ni = x2i /σ
2
i , d
d/2
∏
i
|si| =
∏
i
xi/a
d, distribution (1) can be written in the
following form
w(x) = β exp
{
−adN2
∏
i xi
−
1
2
∑
i
x2i
σ2
i
}
. (7)
In continueous d-dimensional space, general number L of trajectories of random
walk at a given N cannot be determined so simply as in a lattice discrete space. That
is why in continueous space the expression for thermodynamic probability W (x) of
realization of the state, in which the end of chain is on the surface of conformational
ellipsoid with semi-axis x i, should be written in the following way
W (x) = Lw(x). (8)
Here L is a function only of the length of the chain N and Euclidean space
dimension d. If we assume that transitions of the end of the chain in the volume ad
do not create new states we shall receive in the lattice approximation L = (2d)N . In
general, we can assume that order L is determined by the quantity (2d)N .
According to (8) and (7), entropy S(x) = k lnW (x) of this conformational state
can be presented in the following way
S(x) = S0 + Ss(x), (9)
S0 = k lnβL, (10)
Ss(x) = −k
{
adN2
∏
i xi
+
1
2
∑
i
x2i
σ2
i
}
. (11)
The first term S0 in (9) is positive and presents entropy of random walk limited only
by dimension d of space and length of chain N . The second term Ss(x) in (9) makes
negative contribution into entropy of conformation, and the essence of this is that it
presents limitations that cover trajectories of random walk. One of these limitations
is that self-intersection of trajectories of RW is absent, the second one is that the
trajectory must end on the surface of ellipsoid with semi-axis x i, i = 1, d. These are
the limitations which create the effect of self-organization of polymeric chain. That
is why Ss(x) < 0 can be called entropy of self-organization. Thus, conformation of
polymeric chain is a statistic form of its self-organization.
221
Yu.G.Medvedevskikh
Since statistics of SAW does not take into account the effects of short-range order,
the assumption that the free energy of conformation is equal to F (x) = −TS(x) is
natural. Therefore
F (x) = F0 + Fs(x), (12)
F0 = −kT lnβL, (13)
Fs(x) = kT
{
adN2
∏
i xi
+
1
2
∑
i
x2i
σ2
i
}
. (14)
For β ≪ L, the expression F0 ≈ −kTN ln 2d is the free energy of the random
walk. Taking into account the error of “grating” determination of L, the fact that
ln 2d ∼ d/2 and the physical meaning of the quantity F0, we assume
F0 = −
d
2
kTN. (15)
Thus free energy of the random walk is determined by their progressive energy.
Taking into consideration (15) and (14), we shall rewrite (12) in the following
way
F (x) = −
d
2
kTN + kT
{
adN2
∏
i xi
+
1
2
∑
i
x2i
σ2
i
}
. (16)
Among all possible conformational states we shall further consider the most
probable, i.e. the equilibrium ones, which are characterized by the maximum of the
entropy and the minimum of the free energy. Minimizing F (x) under the condition
∂F (x)/∂xi = ∂Fs(x)/∂xi = 0, for the equilibrium state we receive the following
−adN2
/
xi
∏
i
xi + xi/σ
2
i = 0, i = 1, d. (17)
By solving the system of algebraic equations (17), we find equilibrium values of
semi-axis of conformational ellipsoid
xi = σi
(
adN2
∏
i σi
)α
, (18)
where
α = 1/(d+ 2). (19)
We shall particularly underline the states at which all directions of the random
walk of the end of the chain are equilibrium, that is
ni = N/d, i = 1, d. (20)
Then
σ2
i = σ2
0 = a2N, i = 1, d. (21)
222
Thermodynamics of polymeric chains in solution
Substitution of (21) into (18) makes semi-axis xi of the equilibrium ellipsoid equal
to the radius of Flory, xi = Rf , i = 1, d,
Rf = aNν (22)
with Flory index ν = 3α or
ν =
3
(d+ 2)
. (23)
The expression of Flory (22) is a particular case of (18) and is true only at
equal probability of the random walk in all directions of d-dimensional space. If this
condition (20) is not kept equilibrium conformation of polymeric chain acquires the
form of ellipsoid with semi-axes x i, determined by (18). Therefore as a measure of
deformation of polymeric chain that characterizes deviation of its conformational
state from Flory ball it is convenient to take the following variables
ψi = σi/σ0, (24)
u =
∏
i
ψi. (25)
From (6) and (21) it follows that value ψi is covered by the following relation
∑
i
ψ2
i = d. (26)
According to (26) the function u =
∏
i ψi has the maximal value of u = 1 for
Flory ball, when all ψi = 1; at all the other legitimate values of ψi according to (26)
the function u < 1. Consequently any deformation of Flory ball is accompanied by
decrease of u =
∏
i
ψi.
Using (18), we shall express semi-axes xi of equilibrious ellipsoid through param-
eters ψi and u:
xi = Rfψi/u
α. (27)
Having introduced multiplicity of linear deformation in i direction
λi =
xi
Rf
, (28)
we shall receive
∏
i
λi = u2α. (29)
The equation of the relation λi corresponds to the equation of the relation (26)
between λi:
∑
i
λ2i = d
/
∏
i
λi. (30)
We shall receive the expression for free energy F of equilibrium conformation of
polymeric chain by substituting (27) into (16):
F = −
d
2
kTN +
(
1 +
d
2
)
· kT
(
Rf
σ0
)2
/
u2α. (31)
223
Yu.G.Medvedevskikh
For Flory ball we have
Ff = −
d
2
kTN +
(
1 +
d
2
)
· kT
(
Rf
σ0
)2
. (32)
Here
(Rf/σ0)
2 = N (4−d)/(d+2). (33)
As one can see, with the increase of the Flory ball deformation free energy of
equilibrium conformation F increases and it happens at the expense of the free
energy of self-organization Fs:
Fs = (1 + d/2) kT (Rf/σ0)
2 /u2α. (34)
For the Flory ball
Fsf = (1 + d/2)kT (Rf/σ0)
2. (35)
At a given d and N the change of the free energy at the equilibrium deformation
of the Flory ball is determined as a difference ∆F = F − F f = Fs − Fsf . Therefore
according to (34) and (35) we have
∆F = (1 + d/2)kT (Rf/σ0)
2(1/u2α − 1). (36)
At any deformations of Flory ball ∆F > 0, subsequently the work is done with
the system.
For d = 3-dimensional space we have
∆F =
5
6
kTN1/5
(
λ2x + λ2y + λ2z − 3
)
. (37)
Let us compare (37) with the well-known expression of work W of deformation
of caoutchouc volume unit [5]
W =
1
2
kTNc
(
λ2x + λ2y + λ2z − 3
)
. (38)
where Nc is the number of chain per unit of volume.
Similarity of the expressions (37) and (38) is obvious enough. But substantial
differences are more important. The first difference is that in (37) the multiplierN 1/5
is presented. It appears only in RWSA statistics and indicates that as the chain grows
longer the work of Flory ball deformation increases. The second difference is that
because of the presence of the multiplier Nc the work W has dimensions of pressure,
tension, etc., which are the characteristics of elastic properties of caoutchouc. That
is why (38) is also called to be the expression of highly elastic potential of rubber
[6,7]. We shall see later that the elastic properties of polymeric chains are described
by equations different from (38) by their form. Finally we shall note that in the
interpretation of (38) the condition of incompressibility of caoutchouc in the form of
λxλyλz = 1 is used. According to the relation (30), in such a case the expression in
brackets (38) should be equal to 0, i.e., without the change of configuration volume
there is no deformation work.
224
Thermodynamics of polymeric chains in solution
3. The equation of state of equilibrium conformation
Leaving out numeral coefficients, we shall determine the volume V f of Flory ball
in d-dimensional space using of the following expression
Vf = Rd
f , (39)
and the volume V of conformational ellipsoid with the expression
V =
∏
i
xi. (40)
Thus, taking into account (27) we shall receive
V = Vfu
2α = Rd
f u
2α. (41)
So, at any deformations of Flory ball, the volume of configuration decreases.
Having introduced multiplicity of volumetric deformation by the relation λ V =
V/Vf , we have
λv = u2α =
∏
i
λi. (42)
Outer hydrostatic pressure cannot considerably influence the conformational
state of a polymeric chain in the solution. But at deformation of Flory ball, the
volume of the conformation decreases and the free energy increases. Subsequently,
according to thermodynamic relation ∂F/∂V = −P there should exist the measure
of connection between free energy F and volume V of configuration, which plays
the role of conformation pressure. Thus pressure P of conformation is determined
by the following expression
∂F/∂V = ∂Fs/∂V = −P. (43)
Differentiating (31) or (34) and taking into account (41), we receive
P = (1 + d/2)kT (Rf/σ0)
2/Rd
f u
4α. (44)
For Flory ball
Pf = (1 + d/2)kT (Rf/σ0)
2/Rd
f . (45)
Comparing (41) and (44), we shall receive the equation of conformational state
of polymeric chain in the solution
PV 2 = const, (46)
where
const =
(
1 +
d
2
)
kT (Rf/σ0)
2Rd
f . (47)
On the other hand, comparing (41) and (44) with the expression for Fs (34), we
find
Fs = PV (48)
225
Yu.G.Medvedevskikh
and subsequently pressure of conformation P = Fs/V is numerically equal to the
density of free energy of self-organization of polymeric chain.
From (48) taking into account (46) and (47) we have
FsV = const. (49)
Thus, values PV 2 and FsV are integrals of the process of equilibrious deformation
of polymeric chain.
4. Modules of elasticity of polymeric chain
Within the approximation of isotropy of polymeric chain, its relative deformation
∂xi/x in i direction of d-dimensional space under the influence of all main forces f i,
i = 1, d in differential form is described by the expression [9]
Y ∂xi/xi = ∂fi
/
∏
j 6=i
xj + γ
∑
j 6=i
∂fj
/
∏
k 6=j
xk, (50)
where Y is Young’s module, γ is Poisson’s ratio,
∏
j 6=1
xj , and
∏
k 6=j
xk are the values of the
areas in d-dimensional space that are normal to the forces fi and fj correspondingly.
Let us rewrite (50) with regard to Young’s module in the form
Y =
x2i
∏
i xi
∂fi
∂xi
+ γ
∑
j 6=i
xixj
∏
i xi
∂fj
∂xi
. (51)
Equations (50) and (51) are written in the system of signs of mechanics according
to which fi = ∂F (x)/∂xi. Differentiating (16), we have
fi/kT = −adN2
/
xi
∏
i
xi + xi/σ
2
i . (52)
At equilibrium deformation in any current conformational state, the forces should
be equal to 0, and this is expressed by the condition of equilibrium (17). But deriva-
tives ∂fi/∂xi and ∂fj/∂xi as factors of equilibrium process are not equal to zero.
That is why differentiating (52) with respect to xj and xj and substituting into the
resulting expressions equilibrious values x i, i = 1, d by (18) and (27), we have
∂fi/∂xi = 3kT/σ2
0ψ
2
i , (53)
∂fj/∂xi = ∂fi/∂xj = kT/σ2
0ψiψj. (54)
Substitution of (53) and (53) into (51) gives
Y = kT (3 + γ(d− 1)) (Rf/σ0)
2 /Rd
f u
4α. (55)
For Flory blob, Young’s module is equal to
Yf = kT (3 + γ(d− 1)) (Rf/σ0)
2 /Rd
f , (56)
226
Thermodynamics of polymeric chains in solution
that is why (55) can be written in the following form
Y = Yf/u
4α = Yf/λ
2
v. (57)
As one can see from comparison of (44) and (53), Young’s module and pressure
are different only by numerical constants. In case of d = 3-dimensional space, the
connection between Young’s module and pressure is expressed by volume module
E = −V ∂P/∂V using the correlation E = Y/3(1 − 2γ). Taking into account the
logics of its derivation [9], for d-dimensional space we can write
E = Y/d(1− γ(d− 1)). (58)
From the equation of state (46), it follows
E = −V ∂P/∂V = 2P. (59)
Thus,
Y = 2(3 + γ(d− 1))P/(d+ 2). (60)
Using expressions (45) and (58), we shall solve (63) with regard to γ
γ = (d+ 3)/(d+ 1)2. (61)
As one can see, Poisson’s ratio γ depends only upon dimension d of Euclidean space.
Namely, for d = 3, 2 and 1 we have γ = 3/8, 5/9 and 1; accordingly Y = 3/2P ,
16/9P and 2P . As it was expected, in d = 1-dimensional space, Young’s module
coincides with volume module. Through Young’s module and Poisson’s ratio we find
module of shift µ(d > 2) [9]
µ = Y/2(1 + γ). (62)
For d = 3-dimensional space µ = 4/11, Y = 6/11P .
5. Young’s module of caoutchouc
It would be incorrect to make direct theoretic continuation of elastic properties
of polymeric chain in a solution in order to describe elastic properties of caoutchouc.
But we can suggest a semi-empirical approach as in [4,5]. Thus, in the expression
(55) of Young’s module of polymeric chain in a solution instead of blob volume
Rd
f , that is theoretically calculated using Flory formula (22), empirical volume of
polymeric chain in caoutchouc is introduced
1/Rd
f = ρ/MNA, (63)
where ρ is the density of caoutchouc; M is the molecular mass of polymeric chain;
NA is Avogadro number.
With such a substitution, the expression for Young’s module of caoutchouc in
d = 3-dimensional space (γ = 3/8) acquires the following semiempirical form
Yc = 3.75RTN1/5
( ρ
M
) /
u4α. (64)
227
Yu.G.Medvedevskikh
6. Main tensions
The relation between tension Gi in the plane normal to i direction of deforma-
tion and its relative value ∂xi/xi in differential form is described by the following
expression
∂Gi = Y ∂xi/xi. (65)
The analysis shows that (65) has no analytical solution in general form, but
it permits to get the necessary equations of connection. We shall receive the first
equation of connection by substituting Young’s module in the form of (57) and
taking the sum of (65) in all directions of d-dimensional space:
∑
i
Gi = −(Yf/2)
(
1/u4α − 1
)
. (66)
Thus, the sum of main tensions is negative, which is the result of the decrease
of conformational volume of polymeric chain. Taking into account the relation (29),
equation (66) can acquire the following form
∑
i
Gi = − (Yf/2)
(
1
/
∏
i
λ2i − 1
)
. (67)
The second equation of relation between Gi we shall find as follows
∏
i
Gi = (−1)d(Yf/2d)
d
∏
i
(1/λ2di − 1). (68)
At additional suggestions concerning the character of deformation of polymeric
chain simultaneous solution of (67) and (68) enables us to calculate all main tensions
Gi.
7. Forces and work of deformation
We shall define the force of deformation as the force that should be applied
to Flory blob, the radius of which Rf is equilibrium with respect to the values
σi = σ0, i = 1, d, in order to transform it into the state of ellipsoid with semi-axes
xi, equilibrium with regard to values σi 6= σ0, i = 1, d. According to this definition in
expression (53) one should use σi = σ0, i = 1, d, with the values of xi taken according
to (18) as equilibrium with respect to σi 6= σ0. Making necessary substitution in (53),
we shall get the expression for force of deformation
fi = kT
(
Rf/σ
2
0
) (
ψ2
i − 1
)
/ψiu
α. (69)
According to the adopted system of signs fi > 0 at stretch (ψi > 1) and fi < 0
at compression (ψi < 1).
Forces of deformation are not arbitrary, they satisfy the following relations
∑
i
fiψi = 0,
∑
i
fiλi = 0. (70)
228
Thermodynamics of polymeric chains in solution
Work of deformation in all main directions is determined by the expression
A =
∑
i
xi
∫
Rf
fi∂xi. (71)
Substitution of (69) into (71) and taking into account (27) gives
A = kT (Rf/σ0)
2
xi
∫
Rf
(
∑
i
xi∂xi
R2
f
−
∑
i
∂xi
xiu2α
)
. (72)
Having integrated (72) we shall receive
A = (1 + d/2) kT (Rf/σ0)
2 (1/u2α − 1
)
, (73)
that is identical to (36): A = ∆F . This relation proves the correctness of determining
the deformation force in (69).
8. Calculations and illustrations
Let us limit ourselves only to calculations of thermodynamic and elastic proper-
ties of polymeric chain in real d = 3-dimensional Euclidean space. But in this case
let us consider a particular situation, though obviously the most typical one.
Let us assume that among three main forces fx, fy and fz only one, let it be fz,
is an independent variable, i.e. the outside force that influences the polymeric chain.
Then forces fx and fy will be reactions for fz. If fz > 0 and stretch of polymeric
chain takes place along the axis z, then reactions fx and fy are negative and along
the axes x and y polymeric chain compresses. If polymeric chain is isotropic, then
forces and multiplicities of linear deformations along the axes x and y will be equal:
fx = fy, λx = λy. Conformational state of such a polymeric chain has the form of a
prolonged (along the axis z) ellipsoid of rotation. And, on the contrary, if fz < 0 and
polymeric chain compresses along the axis z, then fx = fy > 0 and polymeric chain
stretches along the axes x and y transferring according to the condition λz < 1,
λx = λy > 1 into oblate (along the axis z) ellipsoid of rotation.
For ellipsoid of rotation not only λx = λy, but also ψx = ψy. That is why the
main equations of relations (26) and (30) acquires the following form
2ψ2
x + ψ2
z = 3, (74)
2λ2x + λ2z = 3/λ2xλz. (75)
Here α = 1/5,
u = ψ2
xψz, (76)
λx = ψx/(ψ
2
xψz)
1/5, λz = ψz/(ψ
2
xψz)
1/5, (77)
λv = (ψ2
xψz)
2/5 = λ2xλz. (78)
229
Yu.G.Medvedevskikh
Figure 1. Relationship between param-
eters ψx and ψz of equation (74) dur-
ing the expansion (ψz > 1) and pressure
(ψz < 1) of polymer chain at z axis.
Figure 2. Relationship between de-
grees of linear deformations λx and λz
of equation (75) during the expansion
(λz > 1) and pressure (λz < 1) of poly-
mer chain at z axis.
P ∆F/kT
Figure 3. Pressure P of polymer
chain conformation during its expansion
(λz > 1) and pressure (λz < 1) at z axis.
N = 243, a = 0.186 nm, T = 298 K.
Figure 4. Work A = ∆F of polymer
chain deformation during it expansion
(λz > 1) and pressure (λz < 1) at z
axis. N = 243.
230
Thermodynamics of polymeric chains in solution
P
Figure 5. Young’s module of polymer chains in solution (2 – a = 0.186 nm, 3 –
a = 0.125 nm, calculation as 3P/2) and caoutchouc (1 – calculation by equation
(64), a = 0.125 nm). N = 1024, T = 298 K.
P N
Figure 6. The main exertions Gx and
Gz during the expansion (λz > 1) and
pressure (λz < 1) of polymer chain at
z axis. N = 243, a = 0.186 nm, T =
298 K.
Figure 7. Deformation forces during
the expansion (λz > 1) and pressure
(λz < 1) of polymer chain at z axis.
Calculation by equation (69): N = 243,
a = 0.186 nm, T = 298 K.
231
Yu.G.Medvedevskikh
The relation between parameters ψx and ψz is shown in figure 1, between λx
and λz – in figure 2. The expressions (74)–(77) create the basis for calculating all
thermodynamic and elastic properties of polymeric chain. In figures 3 and 4 the rela-
tionship of pressure of conformation P = 5/2kT/a3N8/5λ2ν and work of deformation
A = ∆F = 5/2kTN1/5(1/λν − 1) is presented. The conditions of calculations are
indicated in cutlines.
Young’s module for polymeric chains in solution is calculated as 3P/2, for caou-
tchouc – according to semi-empirical equation (64). The results of the work are
compared in figure 5.
The equation of relation of (67) and (68) between main tensions Gx and Gz
acquire the form
2Gx +Gz = −
1
2
Yf
(
1/λ2v − 1
)
, (79)
G2
xGz = −(Yf/6)
3(1/λ6x − 1)2(1/λ6z − 1). (80)
Simultaneous solution of right parts of (79) and (80) leads to solving of an
abridged cubic equation.
The results of calculations Gx and Gz are presented in figure 6.
The forces of deformation fx and fy are calculated according to the equation (77)
and illustrated in figure 7. As short comments, we shall note that for three times
stretch of polystyrene molecular in the solution (N = 243) along the axis z (λz = 3)
it is necessary to apply forces of deformation fz ≈ 5 · 10−12 H, which is equal to
gravity force created by mass 5 · 10−13 g. Mass of polystyrene molecular is equal to
∼ 4 · 10−20 g. So, elastic properties of polymeric chain in the solution are extensive
enough.
References
1. Kuhn W. // 1936, vol. 76, p. 258.
2. James H.M., Guth E. // J. Chem. Phys., 1943, vol. 11, p. 455.
3. Treloar L.R.G. // Trans. Faraday Soc., 1943, vol. 39, p. 36; vol. 39, p. 241.
4. Kuhn W. // Koll. Zs., 1934, vol. 68, p. 2.
5. Treloar L. The Physics of Rubber Elasticity. Oxford, 1949.
6. Bartenev G.M., Frenkel C.Ya. Physics of Polymers. Leningrad, Chimiya, 1990 (in
Russian).
7. Askadskiy A.A. Deformation of Polymers. Moscow, Chimiya, 1973 (in Russian).
8. Medvedevskikh Yu.G. // Condens. Matter Phys, 2001, vol. 4, No. 2(26), p. 209.
Statistics of linear polymer chains in the model of random wanderings excluding self-
intersection (in Russian).
9. Feynman R., Leighton R., Sands M. The Feynman Lectures of Physics. vol. 2. Mas-
sachusetts, 1964.
232
Thermodynamics of polymeric chains in solution
Термодинаміка конформації та деформації лінійних
полімерних ланцюгів у розчині
Ю.Г.Медведевських
Відділення Інституту фізичної хімії
ім. Л.В.Писаржевського НАН України
79053 Львів, вул. Наукова, 3а
Отримано 20 грудня 2000 р.
Термодинаміка конформації і деформації лінійних полімерних лан-
цюгів у розчині побудована на основі статистики блукань без пере-
хрещення. Остання виражає ентропію та вільну енергію конформа-
ції полімерного ланцюга у вигляді суми доданків, один з яких вра-
ховує внесок випадкових блукань, інший – двох обмежень, що на-
кладаються на випадкові блукання і створюють ефект самоорганіза-
ції полімерного ланцюга. Його деформація розглянута у вигляді рів-
новажного переходу клубка Флорі в конформаційний елiпсоїд. За-
пропоновані вирази термодинамічних та пружних властивостей по-
лімерного ланцюга як функцій ступеня його деформації. Об’ємний
модуль, модулі Юнга та зсуву полімерного ланцюга виражаються че-
рез тиск конформації, коефіцієнт Пуассона залежить тільки від роз-
мірності евклідового простору. Визначені сили та робота деформа-
ції, запропонований метод розрахунку головних напружень.
Ключові слова: конформація, полімерний ланцюг, термодинаміка,
деформація, пружні властивості, сила, робота
PACS: 05.70.Ce
233
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