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Entanglement of Thread Molecules and Its Influence on the Properties of Polymers.

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Entanglement of Thread Molecules and Its Influence on the Properties
of Polymers
By Martin HoffmannL"]
Dedicated t o Professor Otto Bayer on his 75th birthday
This article shows the extent to which the entanglement of threadlike molecules can influence
many of the properties of deformable polymers and of solutions of macromolecular substances.
Thus, the molecular concept of entanglement leads to quantitative predictions of the dependence
of the entanglement number on the nature of the polymer, its concentration, and extension.
Experimental values of the relaxation modulus confirm these predictions for melts and solutions.
The influence of entanglement on relaxation and flow processes leads to very simple relationships
for the magnitudes of the structural viscosity and the shear stress at which the structural
viscosity starts to become measurable. Osmotic pressures, light scattering intensities, and diffusion
coefficients of concentrated solutions can be derived quantitatively from the conformational
constraints due to entanglement. Entanglement is effective above a concentration at which
the three-dimensional coils fill the solution volume, and probably also at lower concentrations.
At very high concentrations entangled structures may be formed which drastically reduce
the number of possible conformations. The swelling and extension behavior of cross-linked
deformable polymers can be understood only if entanglement is taken into account.
1. Introduction
In dilute solutions['] and in melts[21long threadlike molecules demonstrably assume a conformation resembling a threedimensional coil: for a polystyrene sample of molecular weight
M = 1 x lo6 in toluene solution the diameter of the coil is
about 100nm. Such coils are approximately spherical and
in this case have a volume V of about 3 x lo8ml per mole.
The polymer concentration in the coil is M/Vand here amounts
to about 0.3 wt-%. In solutions containing more than 0.3
wt-% of such a polymer the coils may interpenetrate or escape
interpenetration by shrinkage. The coils are energetically preferred, i. e. stable, conformations; they rapidly return to their
original shape after deformation (rubber-elasti~ity[~]).
As a
result, they shrink only little on concentrating the solutions,
and they are so large in melts that, with such threadlike
molecules and because of the high polymer concentration
in the melt, the number of molecules contained in the volume
of one coil must be of the order of lo2.If several coils interpenetrate with retention of the coil conformation, then chain fragments must touch one another during the continual movement
of the chains (conformational changes); in some cases the
movement is such that one chain twines round the other
so that further movement is hindered. Such contact between
two chains acts briefly as a chain junction made by main
valences.
This concept of physical cross-linking by entanglement was
first formulated several decades ago (for a review see C41),
but was accepted only slowly as a basis for theoretical interpretation of the properties of polymers and their solutions[4- 61.
Such conformational interaction of polymer molecules does
not arise from any specific interaction by secondary valence
forces between them and therefore occurs generally in solu-
tions and melts, as well as in partly crystalline and crosslinked polymers. Since entanglement influences many of the
properties of high-molecular substances and their solutions,
the underlying principles and effects will now be surveyed
and formulated quantitatively for noncrystalline systems.
2. The Molecular Model
The shape of long molecules essentially resembles a random
flight path, being formed by joining rigid chain fragments
of given length (segments) to one another in such a way
that the direction of the last segment is completely independent
of the direction of the previous one and all directions occur
with equal probability[']. These segments of length / replace
equally long chain fragments regarded as rigid (straight or
possibly curved). The distance h between the ends of a sufficiently long chain of Z segments is then on average a vector
of length $6.
If Z is large, the chain can be divided into
subchains each with Zimembers, and each subchain obeys
chain statistics ( i . e. hi=
The vectors hi at the junctions
are on average perpendicular to one another, as is easily
appreciated by considering the case of Z=2Zi; here h =
and hj=11/2/2, so that h=1/2.hi is the diagonal of a square
of side length hi.
For the description below, which is correct to an order
of magnitude, the segment model can be simplified to the
string-of-beads model; thus, we assume that the segments
are compact balls that in a melt adopt a close packing and
become deformed so as to eliminate empty spaces. The segment
molecular weight M s can then be employed to calculate how
many segments are contained in 1 cm3 of the melt of density
p ( N A z 6 x lo2'));
fi).
fi
[*] Dr. M. Hoffmann
Bayer AG, Zentrale Forschung, Wissenschaftliches Hauptlabor
5090 Leverkusen (Germany)
Angew. Chem. Int. Ed. Engl. 16.751-766 ( l Y 7 7 )
751
O n estimating how many bonds are required for the direction of a polymethylene chain to be continued with equal
probability in eaEh spatial direction (a requirement of the
segment model), one obtains a segment molecular weight M s
of around 170, in agreement with estimates from measured
coil dimensionsr8'; thus for polyethylene Ns is approximately
3 x 1021.
Next, we can calculate how many contacts between different
chains occur in 1 cm3 of this melt. Figure 1 shows two macromolecules A and B in a melt, the two chains being in contact
at segment a of chain A. The segment a has twelve nearest
neighbor sites, of which two are occupied by neighboring
portions of chain A. In addition, back-coiling of chain A
causes further p nearest neighbor sites of segment a to be
occupied by chain A.
2
Fig. 1. Two thread molecules A and B whose spherical segments in the
melt adopt close packing (represented two-dimensionally).
When estimating the back-coiling number p , it must be
remembered that the third segment (counting from a) can
occupy, out of the twelve possible places, those four that
are adjacent to a, so that 4/12=0.33 neighboring sites of a
are occupied. This treatment can be continued for further
segments, considering then the positions of the third segment.
The residence probability P of a more distant segment at
a site adjacent to a can be calculated by the known statistical
function["J for chain-end separations (ais F b r y ' s expansion
factor["]):
(3)
For polyethylene therefore, in conjunction with eq. (I),
N K = 5 x 102'.
It still remains to be shown which of these contacts should
be regarded as entanglements. Entanglements should be contacts that withstand many changes of chain conformation
and thus are so long-lived that they act as cross-linking sites,
permitting rubber-elastic deformations of the melts. Figure
2 shows such a contact position of chains A and B. In this
presentation['] the chains are symbolized only by the vectors
hi of the subchains, which here have the designations A l ,
Fig. 2. Two thread molecules A and B touch at one point. Their subchains
are symbolized by the vectors A , and E l .The vectors hA and he join the
end points of two chain fragments that are themselves situated at further
entanglement points 191.
=
For I= 1 each of the twelve neighboring sites has the volume
d x d y d z = 1 and in eq. (2) the exponent -3/(2Za2). The summation over all values of 12P(Zi), leading to p, should already
be stopped, however, when Zi is still small: entanglements
are continually being untied and new ones formed; the network
chains of the entanglement network, i.e. the portions of the
chain between entanglement points, contain in the melt relatively few segments [see eq. (17)], of which certainly only a
small proportion can be attached at segment a without severely
straining the residual part of the network chain. Contact
of chain A at segment a with distant segments of the same
chain should therefore be regarded as contact between different
chains. In solutions the network chains comprise many more
segments than in melts, so that p can then be greater than
in melts; p z 1 has been estimated for the latter.
752
+
3.5
NK=-N~
A
P(x,y.z)dxdydz
If one considers that chain A coils back to segment a
from two directions, one finds that, of the twelve neighboring
sites of segment a in the melt, 2 + 2 p x 4 sites are occupied
by segments of the neighboring portions of chain A. The
remaining 12-4=8 sites adjacent to segment a can now
be occupied by segments of other chains. If chain B occupies
such a position with segment b, then a and b have four
nearest neighbors in common, of which on average 4 x 4/12
are occupied by chain A and the same number by chain B.
Chain B thus occupies 1 (4/3) = 2.3 of the sites adjacent to a.
The eight sites at a which are free for contact with other
chains can thus be occupied by 8/2.3=3.5 foreign chains.
On counting all NK contacts of the various chains in 1 cm3
ofthe melt, each contact refers to two segments and is therefore
counted twice, giving the contact number NK asr9]:
Azr B1, and B2. In the outer part of Figure 2, these
vectors end in further entanglements of the chains A and
B. Had the contact of chains A and B taken the form shown
in Figure I , it would have been rapidly broken by the usual
conformational changes. On average, such changes leave unaltered the length of the vectors A l , Az, B1, BZ and angle
CI (on average a right angle) between Al and A, and the
angle p between B1 and B2. Having fixed the lengths of the
vectors A, and Bi,as well as a and p, the allowed conformational
changes thus correspond to rotating chain A (or B) around
the vector hA (or h g ) . It is now simple to formulate the conditions under which contact between chains A and B is not
destroyed by such rotations. In so doing, it must be borne
in mind that h A and h g can assume any position in space.
The contact is not destroyed if the following geometric conditions are fulfilled (for a =/.'=90"):
a) Chain B touches chain A in the internal angle of the
vectors A , and A2. This happens with a probability W,= 114
(=90"/360").
Angrw. Chem. I n t . Ed. Engl. 16.751-766 ( 1 9 7 7 )
b) Vector B1 lies above and vector B2 below the plane
of the vector A l and A 2 : Wb= 1/2.
c) In cases where the plane containing A, is at right angles
to that of B, : B points in directions in space that correspond
to the octant lying diagonally "opposite" the internal angle
between A l and A 2 : Wc=1/4. W, is not changed when the
plane of A, is not at a right angle to that of B,.
The fraction W= W, x Wb x W,=O.O31 of all NKcontacts is
therefore long-lived. Such contacts are entanglements that are
only destroyed when neighboring contacts are also released
in a cooperative process.
This argument can be extended to all cases in which the
angles x and/or are not 90".
If/1=90", then W2 is always 1/2 and Wi =z/360, and further
W,=(180-~()/360. The probability W,, then amounts to:
8
[M( 180- .)./I(180
3604
W/j = ----
-/I)]
(4)
/3 can assume any values between 0 and 180°, so that
the probabilities are proportional to sincc and sinj?, respectively, the maximum of the frequency distribution of statistical
conformations lies at 90°, and half of all cases have angles
between 60 and 120". If Wziiis multiplied by the corresponding
values of the frequencies of x and and averaged, the result
is WR=2/n4=0.021. Furthermore, W, is not changed when
the vectors A], A2, B 1 , B2 are allowed to have different lengths.
If, however, the entanglements are sufficiently long-lived, then
Ai and B, "forget" to which chain they belong and thus on
average point to the corners of a tetrahedron. The frequency
distribution of the angles then has a maximum at 109.5",
so that the probability wfalls to ~10,,5=0.018[91.
The entanglement number N , is then:
to this end faster than a more distant oner9].Such considerations will be taken up in Section 4 below.
We would add here that, even in the absence of any long-term
relaxation, the entanglement number N , given by eq. (5) is
lowered if the entanglement network is subjected to shear
or deformation['2]. Deformation causes changes in the angles
z and /3 and thus in the geometric requirements for entangle2
/
/
I)/-
- -X
z and
N , =0.01a N~
(51
For polymethylene eqs. (4) and (3) give N,,=9 x
The preceding derivation neglects the effect of end groups
and thus assumes infinitely long molecules. If the network
chains of the entanglement network have a molecular weight
M, and the molecules have a molecular weight M , then
networks can be
only if the condition known for
cross-linking polymerizations[' '1 is fulfilled for the construction of a continuous network:
In the derivation of eq. (5) it was also considered unimportant that some of such entanglements are more easily broken
than others. Thus, the entanglement number calculated according to eq. (5)includes entanglement for which the pair of values
z and / j or the position of chain A relative to chain B only
just fulfil the geometric requirements for entanglement. Even
small diffusion movements of neighboring entanglement points
can break such entanglements, so that they will certainly
be shorter-lived than the others.
Furthermore, n o consideration was given to how easily
the chains could slide over one another and to how far the
end of the chain is separated from the entanglement of interest.
Diffusion of a chain end could break an entanglement near
Anyew. Chem. Int. Ed. Engl. 16, 751-766 ( 1 9 7 7 )
.
Fig. 3. As a result of stretching the spatial direct~onsand lengths of the
vectors at a (tetrahedral) contact of two chains are deformed, so that the
entanglement no longer has the same probabllity W7/j [see Eq. (4)] as before
the stretching [12].
ment. The tetrahedral entanglement (Fig. 3) is broken if suficiently strong uniaxial stretching occurs in directions that
do not lie in the part of the spherical shell drawn in the
figure (or in the opposite direction), for then the stretching
directions make an angle smaller than 90" with one or more
of the relevant chain vectors. Thus, purely geometrically and
in a very short time, a large stretch reduces the number
of entanglement points per unit volume to the fraction X :
(7)
If one averages over all x and /j corresponding to their frequency, one obtains['21for a maximum of the frequency distribution 90" X=O.33 and for 109.5" an almost identical value.
If one now permits even a slight relaxation or slip of the
sometimes very unevenly stretched network chains at the
entanglement points, then the number of residual entanglement
points decreases almost to zero.
Furthermore, it is important to know how small extensions
reduce the entanglement number N,. The ends of the vectors
A', A 2 , B1,and B2 can occupy any desired point on the
surface of a sphere of radius IAl. On deformation (extension
ratio 2 = L/Lo)such a sphere is transformed into an ellipsoid
of rotation ( a = I A l . i , h=IAIfl.), so that on average each
vector assumes a greater length IA,1[31:
153
The coordinate x (in the stretching direction) and y of each
vectorend-point have thuschanged i n t o x ' = x . i and y ' = y f i
so that an angle y formed initially by the x-axis and such
a vector (tany=y/x) changes to ;*' (tany'=y'/~'=tany/i'.~).
If one puts the length 1Aj.l into the equation for the ellipsoid
of rotation one obtains the relevant angle x' between two
vectors as tan(c('/2)= f1/3.'.'. Before the stretching, the angles
a and amount on average to 90" and are transformed into
smaller or greater angles N' and p'. As a result, WxP[eq.
(4)] decreases to Wx.lI.,
i. e. the entanglement number N , falls
to N:. The factorf;. (= W g o , g o , j / W ~ ~ gives
, ~ ~ , the
; . ~fraction
l)
of entanglements that are still present at i. (relative to the
number at ;"=I) (Fig. 4a). The elastic restoring force of an
entanglement network (referred to the cross section of the
stretched specimen) is therefore not, as in main-valence
networks, proportional to 2' - 1/2 [see eq. (9)], but is instead
proportional to f,.(;.* - I/>.). Figure 4 b shows the course of
this function. Above a certain extension the restoring force
should decrease on further extension. The entanglement
network thus breaks on deformations around 1= 2. The experimentally observed extension Law for non-cross-linked, very
slowly relaxing polymers confirms that the entanglement
number N , decreases with 2 in the manner derived here[l21.
Moreover, the shear stress at which structural viscosity occurs
shows that entanglement networks break at specific extensions
(see Section 5).
that, although a neighbor along the chain, is not the junction
nearest in space but in fact one further removed (Fig. 5)
(see also Section 8). If such a network is stretched, volume
elements of interest become more remote in the stretching
direction.
b
R
Fig. 5. At high concentrations the distances between the entanglement points
are smaller than those between the ends of the network chains. Then several
simple networks seem to interpenetrate (different point symbols for each
network). If such a system is deformed, contacts are established that were
not previously present; these contacts are not broken during extension, thus
acting as additional entanglement points [IZ].
Since the contour length of the network chains remains
unchanged, the network chain finally contacts a spatially close
network junction["' (Fig. 5). Even a small extension can lead
in this way to larger entanglement numbers, estimated["]
as (S/3)N , , = 1 . After a small extension, therefore, the N , , =
in eq. (5) should be replaced by (S/3)N,,,=
3. Conclusions from the Molecular Model and Their
Experimental Testing with the Aid of the Relaxation
Modulus
Equation (5), the result of the molecular model, can be
tested by investigating whether melts have highly elastic properties. As shown in Figure 6, it is in fact found experimentally
that above the softening temperature a rapid deformation
is followed by restoring forces cr that remain constant for
a longer time (plateau of the relaxation modulus in the rubber
region of the relaxation). Use of the extension ratio i= L/Lo
permits calculation of the relaxation moduli E, or G,=3E,
(G,=shear modulus in experiments withf;.z
I
7
2
x-
3
(9)
--..
2
x-
.- ..
3
Fig. 4. a) The percentage of active entanglements as a function of the extension
ratioi.f,.= W90,so,;./W90.90,;.=, is not exactly I/;. as required by the MooneyRivlin equation [19]. b) The restoring force ur of an entanglement network
divided by the relaxation modulus G , (at .; = 1) in dependence on the extension
ratio i. [cf. eq. (9)]. If iis larger than corresponds to the curve maximum,
an equilibrium retractive force is no longer established: the sample flows
by a mechanism different to the usual one, where the conformational changes
destroy and rebuild entanglements even at low T, The relationship applies
for all polymers.
The above considerations neglect the fact that in entanglement networks of undiluted polymers (and in very concentrated solutions) the network chains lead to a network junction
IS4
At sufficiently high molecular weights G, is proportional to
2N,, the number of network chains per cm3 of the melt,
and it has a finite value only when the molecular weight
M , ( M , = number-average molecular weight of the polymer)
of the polymer exceeds 2 M , (Fig. 6).
However, at appreciably larger M , G, is independent of
M . This result is to be expected from the description of entanglement given above, because at M < 2 M , the known requirement for network formation [see eq. (6)] is not fulfilled. On
the other hand, if 2 M , e M , the entanglement number does
not depend on the molecular weight because the small number
of chain ends does not influence the relaxation modulus.
For polyethylene melts at 120°C one obtains according
to eq. (9) a modulus G, of about 1.1 x 106Pa, and thus an
entanglement number N , = 8 x 10' at 25 "C. The agreement
of this value with that estimated from eq. (4) is much too
good in view of the uncertainty in the segment size, the crude
approximations of the molecular model used here, and the
uncertainty in the conversion from 120 to 25°C. The experiment thus confirms the molecular model at least t o within
Angew. Chem. Int. Ed. Engl. 16,751-766 ( 1 9 7 7 )
10’
an order of magnitude and encourages us in our search for
further conclusions.
In proceeding further it must be remembered that the segment model correctly represents the conformations of a chain
of CH2 groups linked together at the usual valence angles. The
auxiliary quantity “segment” is used only to simplify the statistics. In reality the spheres considered here are the CH2 groups
of the polymethylene chain and these are joined together
with specific valence angles. If to a polymethylene melt we
add oligomers or solvent molecules having the size of one
segment (or the size of the CH2 groups), not changing the
chain conformation by special interactions with the polymer,
then the solvent molecules will occupy some neighboring sites
of each segment and thus reduce the number of contacts
between different chains (contact number NK).The probability
of pairs of segments from different chains occurring is known
to be proportional to the square of the volume fraction cp
of the polymer, provided that the back-coiling number p is
not dependent on cp:
*\
\
lo8
\
\
\
\
\
-t
\
-
lo7
2
lo6
v
ae.
\
\
- - --
10’
- strong cross-linking
weak cross-linking
--
10‘
M = 0.5 1.5
5 15.105
NK=
, ~N K , .‘p2
I
ts1Fig. 6. Restoring forces ur arise after momentary stretching of cis-1,4-polyisoprene, which decrease as the load-application time is increased. In non-crosslinked samples of molecular weight M > 2 x lo4 it is observed that G , (the
relaxation modulus calculated from the restoring force) remains nearly constant for a certain time (the plateau of the modulus in the rubber elastic
region of the relaxation). G, is there almost independent of the molecular
weight and of weak cross-linking by main valences [14].
(10)
t
Since, furthermore, the geometric conditions of chain conformation at entanglement points are not changed by the presence
of solvent molecules, it follows from eq. (10) and (5) that
the entanglement number is[’]:
N,,,= Nv31 .‘pz
(1 1)
Table 1. Relaxation moduli G , , critical shear stresses T* (see text), molecular weights for network chains M,, and molar volumes V, of the repeating unit
for the ciitanglenient network:, i n linear polymers, in dependence o n thc volume fraction q and temperature.
Polymei
Polymethylene
Polybutadiene (Li)
Polyisoprene (cis-1,4)
Polyisobutylene
Polydimethylsiloxane
Poly(viny1 acetate)
Polystyrene
Polyibutyl acrylate)
Poly(2-ethylhexyl acrylate)
Poly(dodecy1 methacrylate)
Polyisoprene (Li)
(M=1.7~
10’)
in toluene
Polystyrene
( M . = 6 x 106. c‘, :
0.3)
iii
toluene
in decalin
Polystyrene
( M = I . l 5 x 10‘)
in toluene
in decalin
Angew. Chum. Int. Ed. Engl. 16,751-766 ( 1 9 7 7 )
+ 25
i
25
+ 25
+ 25
f 25
+25
+25
i 25
i
25
25
-51
- 22
i25
84
25
25
i 25
+ 25
125
25
i25
25
+25
25
25
25
25
25
25
i 25
i 25
i 25
i 25
+
+
+
+
+
+
+
+
+
+
+
+
+ 25
+ 25
31
32
37
62
72
73
91
129
203
285
37
91
97
91
1
1
1
1
1
1
1
1
I
1
1
1
1
1
0.59
0.44
0.32
0.26
0.20
0.19
0.17
0.11
0.055
0.21
0 13
0.12
0.059
0.028
0.12
0.058
0.028
0.25
0.12
0.2s
0.12
90
-..
55
:
70
32
20
10
10
5.5
4.0
3.0
1.7
28
29
30
29
10.5
1.9
3.5
1.45
0.70
0.90
0.55
0.13
0.012
0.56
0.28
0.24
0.055
0.006
0.16
0.035
0.005
0.76
0.06
:
30
0 75
0.050
5 30
11.0
8.0
9.0
3.0
3.0
I .I
2 500
4 300
7 300
I I 000
25000
30000
45 000
65 000
ROOOO
140000
17
5.3
I .5
0.4
0.15
0.01 5
0.40
0.15
0.1 5
0.045
0.008
0.14
0.03
0.70
0.14
0.60
0.14
155
Table 1 shows that the relaxation modulus G,, which according to eq. (9)is a measure of the entanglement number N,, does
in fact depend on the square of the volume fraction cp of
the high-molecular components in solutions and polymer mixtures[’]. The same result can be derived from the right-hand
side of Figure 7 K-G,). Fractions with molecular weights
M<M,,,=, thus lower the entanglement density of a polymer.
1.0
1
0.5
1
-
10.1m
-2 0.05
-
0
O.O1I
0.005
.
15 %
5
v,
0.001
t
lo-’
100
10’
-
lo2
t [SI
10’
Fig. 7. Restoring forces arising after momentary (5 ms) deformation of mixtures
oftwo molecularly homogeneous (Uy<0.02) polyisoprenes (Li) ( M I= 2 x 10‘.
M 2 = 1 . 4 x to5) plotted as a function of the load-application time at 22°C.
f;fa has the following meaning: f=u/;, is the restoring force referred t o
the initial cross-section of the sample, and j . is the same for all samples
(3.0bar=Cr; eq. (9)). According to eq. (9), ur-Cp The components relax
at different rates, which leads to the stepwise nature of the relaxation curves
of such mixtures [9]. The numbers on the curves are the proportions of
M I in the mixtures.
When we next consider melts of chains whose side groups
are so large that they occupy neighboring sites of a segment,
and thus decrease the entanglement number N,, we see that,
as in eq. (II), the square of the volume fraction of backbone
chains determines the entanglement number. If V,,,, is the
molar volume of an ethylene unit and V, that of the repeating
unit of a vinyl polymer with side groups, then we should
find that[’]:
Figure 8 and Table 1 show that eq. (12) is also confirmed
by experiment[’]. In these calculations we used the molar
volume V p of the repeating unit of the polymers or the molar
volume of a chain fragment of constant length, this fragment
corresponding to a repeating unit in the case of vinyl polymers.
In eq. (12) it is assumed that the backcoiling number p does
not depend on the structure of the polymer; special hindrance
to rotation is also disregarded.
In place of the number N , of entanglement points in 1 cm3
we may use M,, the molecular weight of the network chains
in the entanglement network (p=density of the polymer).
From equations (9), (1 I), and (12):
-
50 100
10
[rnl.rnol-’]
500
Fig. 8. Relaxation moduli C,, measured according to Fig. 6, of various
polymers (for individual values see Table 1) plotted as a function of the
molar volume of a chain fragment whose length corresponds t o a repeating
unit of the vinyl polymers [9].
A proportionality between M c , , and l/cp was earlier derived
by Bueche by a different route and was tested e~perimentally[~].
At this point we should turn to yet another consequence
of the molecular model, concerning the segment size. Eq.
(3) shows the relation between the segment number N s and
the contact number NKin chains without side groups. According to eq. (12) side groups reduce the contact number proportionally to (K,pE/VE)2.It is now possible to define an apparent
segment size by assuming spherical segments (larger than
in polyethylene because the spheres now contain the side
groups) and apply eq. (3) for the calculation of the segment
number. The segment number in melts is then given by:
It follows that the molecular weight of the segment Ms= & p
in melts is:
Experimentally, it has been found that eq. (16) is confirmed,
although the factor 0.36 is better replaced by 0.2[91.
Now, since according to eq. (13) the molecular weight Mc,t
of the network chain in the melt is also proportional to
V:, the network chains of different polymers in the melt all
contain the same number of segments[’l:
M ~ =,-~
~ 4.1
__
= 11 to 21
MS,,
0.36 to 0.2
For nonideal solutions eq. (14) must be modified when the
distance h between the ends of a molecule in such solutions
is greater by the Flory expansion factor[131Y. than in ideal
756
Angrw. Chrm. I n t . Ed. Engl. 16. 751-766 ( 1 9 7 7 1
solutions (denoted by a subscript 0): h=he.s(. The volume
occupied by a chain fragment of molecular weight M , in
the form of a coil is then increased by the factor ctf over
that in ideal solvents, so that correspondingly more foreign
chains pass through this coil and form entanglements. Moreover, in each volume element there are s(," as many chains,
because at a given total concentration the more coils must
overlap, i. e. interpenetrate, the greater the coils are at a given
molecular weight, so that the total concentration may everywhere be attained. Finally, the segment length also increases,
under the usual assumptions ( Z o16aZ= Z 'I and Z olo = Z 1)
by the factor a:. The segment molecular weight M s thus
similarly increases by about 12, so that N K and hence N ,
should decrease by zf. In all, z, thus changes eq. (14) to['41:
The experimental relaxation moduli confirm (see Table 1 and
Figs. 9 and 19) the influence of the efficiency of the solvent
and thus the influence of x,.
aoot
0.01
9-
0s
1.0
Fig. 9. Relaxation moduli Gr.wdivided by the concentration
(=volume
fractionjplotted as afunction of theconcentration for mixtures of polyisoprene
(Cariflex IR 305, M , , = 1 . 7 x lob, U,<O.I) and toluene, as measured from
the relaxation behavior IS -3000ms after momentary (5 ms) compression
o r shear and rxtrapolatcd to t = O (Hottinger technique, display oscillographj.
The osmotic presrures pox, also plotted were measured with an automatically
compensating osmomeler [I41 (left-hand branch of the curves) or taken
from the literature [29].
According to experimental investigations of structural viscosity["] and relaxation modulus (see Table 1) in a temperature
interval of about I O O T , the temperature dependence of Mc,l
is negligibly small. The modulus plateau can therefore hardly
Angew. C h m lnr. Ed. Enyl. 16, 751 766 ( l Y 7 7 )
be due to an equilibrium between association and dissociation
of the polymer
4. Influences of Entanglement on Relaxation and Flow
Processes
All the investigations described so far show that the entanglement number is an intrinsic property of each polymer and
is thus independent of time. Since, however, according to
Figure 7 restoring forces disappear in the course of time
and melts or solutions can flow, it must be assumed that
the entanglement numbers measured correspond to a dynamic
equilibrium between disentanglement and renewed entanglement. The underlying processes are not specific associations
and dissociations of the polymer chains but the usual conformational changes of the chains, diffusion of chain ends through
entanglement loops[6], and cooperative and nearly simultaneous conformational changes at several neighboring
entanglement p o i n t ~ [ ~ -It~is] .found that polystyrene powder
( M w zlo'), produced by spraying of dilute solutions into
vacuum, has the usual flow behavior immediately after pressing
in vacuum, i.e. about 40min after the spraying. Thus. the
entanglement network must have formed in the course of
this time. Other experiments show that large dissolved molecules diffuse away from one another in milliseconds after
instantaneous fission; then the fragments display the usual
light scattering" 'I. The conformations clearly change so fast
that even complicated structures such as entanglement
networks are formed sufficiently rapidly. Further information
on this is provided by a study of strain relaxation and of flow.
Figure 7 shows relaxation measurements made on mixtures
of polyisoprenes with a heterogeneity index of about 1.1.
It is evident that these mixtures form two kinds of entanglement
networks. At short times one finds moduli that correspond
to the usual moduli for polyisoprenes (Fig. 6), independently
of the composition of the mixture or its molecular weight
M , (weight average molecular weight). This means that the
two types of molecules have formed a common network whose
network density is independent of the composition of the
mixture. At long times, however, there is a second plateau
of the relaxation moduli that shows much smaller moduli
and depends strongly on the concentration of the high-molecular fraction and can therefore be assigned only to entanglement
of the larger molecules. At this time all the entanglement
positions in which short molecules take part have changed
their conformation, their contribution to the restoring force
being lost because the chains have formed an equal number
of new entanglements with the use of undeformed chain fragments. The entanglements of the larger molecules remain;
these are broken only after longer times, because there must
be a cooperative process involving many entanglement points
simultaneously-or at least coordinated in time and spaceand this cooperation requires the more points, the larger
is the molecule concerned.
Buechd'] and others (for a review see l41) have used the
concept of entanglement in theories of flow. Accordingly, the
maximal relaxation time t , in the rubber-elasticity region of
the relaxation and the viscosity q proportional to it depend
extensively on the molecular weight M and the concentration
cp (volume fraction)" '1:
t , =const.tT =const.
kf3.'.(oJ
(20)
751
Figure 7 gives the times in which the entanglement networks
of molecules of various sizes undergo strain relaxation to
one-tenth of its original value; these times are approximately
in the ratio (M2/M1)3.5,
thus confirming eq. (20). Furthermore,
the step heights obey approximately the expected law, namely
eq. (13). On the other hand, concentration dependence of
t , in this experiment appears to be considerably weaker than
predicted by eq. (20); the reason for this is that the process
cannot be described by a single relaxation time, and the maximal relaxation times t , for disappearance of the last traces
of a restoring force cannot yet be measured with accuracy.
However, many investigations of the flow of solutions confirm the concentration dependence expressed in eq. (20) (Figs.
10 and 11). Naturally, this is only so if the prerequisites
for eq. (20) are satisfied, i.e. if the expansion factor x and
the back-coiling number p as well as the segment mobility
are none of them strongly dependent on ( P [ ~ . ' ~ I .
The flow behavior of melts and solutions containing
entangled molecules also shows other effects of entanglement.
It is found that the molecular volume V, of the repeating
unit of the vinyl polymers has an appreciable influence[' 71:
The factor f ( T- T,) describes the influence of the strongly
temperature-dependent segment mobility. In the absence of
entanglement the viscosity is far less influenced by M , cp,
and &["I:
I .
. .
10
1
c [g /lo0 rnll-
01
Fig. 10. Relative viscosity q/qtOl (from flow curves extrapolated to shear
stress T =0) plotted as a function of the concentration for polystyrene fractions
of various molecular weights. [q] =limiting viscosity in [dl/g] in toluene
at 25°C. Above a critical concentration the viscosity of the solutions increases
with c4. The critical concentration decreases with increasing molecular weight.
Thus, eq. (21) holds for the dependence of the viscosity on
molecular weight at molecular weights M that fulfil the
network-formation condition [eq. (6)], whereas eq. (22) applies
if M is smaller. This kink or transition in the logq-logM
relationship has been shown to appear in many polymers
and solution^[^^ ', 7 32 1 1 .
M.,
4.3.106
i
10:
0
10'
6
F
'I
g,
-+/-+
P'
-1
10.
0
a
m
F
10'
/'
4
1I =5570
'
10'
2
10'
(ED2
MI3
M)5
01
Q2
0.3
45
cpFig. I I . Flow curves for mixtures of two polyisoprenes (Li, M 1 = 1 . 4 x lo5
or 1.2 x 10'; M 2 = 2 x [ O h , UM<O.l)give viscosities q i = o and q T = x extrapolated t o shear stresses r=O and T = J ,respectively, which are here plotted
as a function of the concentration (=volume fraction m) [17]. The h = z .
is extrapolated as if the viscosity q 1 of the lower-molecular component were
independent Of T and at sufficiently small q , is proportion t o 'p2.At Sufficiently
small q , , however, ~ i is ~ proportional
= ~
to q4. q l = l . l x lo5 refers to
M 1 = 1 . 4 x 1 0 5 ; ~ 1 1 = 5 . 510'
x refers t o M I = l . 2 x 1 0 4 .
758
103
lo4
T [Pal+
Fig. 12. Flow curves of unbranched polydodecyl methacrylates at 25°C.
The flow curves of these polymers can be measured down to the vicinity
of the second Newtonian region without fracture of the melt [17].
Anyrw. Chem. l n t . Ed. Engl. 16,751-766 ( 1 9 7 7 )
A further consequence of entanglement is the appearance
of a structural (non-Newtonian) viscosity that is much stronger
than in systems where the thread molecules d o not entangle.
If melts or solutions that follow eq. (21) are made to flow
under higher shear forces, their viscosity decreases with
increasing shear stress 7 (Fig. 12). At very high shear stresses
the viscosity reaches values that correspond to those of eq.
(22)[17],so long as the melt does not previously collapse (melt
fracture). The quotient of the viscosities qv and q constitutes
a suitable measure of the magnitude of the structural viscosity:
Figure 13 shows flow curves for polyisoprene mixtures[g1at
low shear stresses. Measurements on solutions["] and on
the mixtures of Figure 13 are shown in Figure 14 to conform,
at least approximately, to eq. (23) which, as previously, is
formulated for constant expansion factors a, and constant
back-coiling numbers p . The influence of the basic molar
volume V, is not clearly visible for the few polymers studied,
lo5-
I
I
10' J
0I
0.01
1.0
v Fig. 14. q r = ~ / q r = r as a measure of the magnitude of the structural viscosity
plotted against the concentration for polystyrenes ( M = 1.4 x lo', Uhc10.3)
in toluene (o), and for polyisoprene ( M = 1.7 x lo", U M s O . l )in polyisoprenes
of lower molecular weight ( 0 ; see Fig. 13) as well as in toluene ( + ) at
25°C. The concentration dependence sufficiently confirms eq. (23).
To obtain a visible structural viscosity of the melts it is
necessary to use various shear stresses, suited to the individual
polymer. For a more precise discussion we use the shear
stress z, found at the inflection point of the plots of logq
uersus logt. Figure 12 shows that the z, are almost independent
ofM, so that zwmay be regarded as a property of entanglement
networks. Furthermore, the r, follow laws similar to those
for Grr9, 'I. Eq. (24) summarizes the results for several polymers
at various concentrations (see also Table 1):
r
10'
I
lo4
t [Pal-
lo5
Fig. 13. Flow curves of mixtures of homogeneous ( U ~ c 0 . 1 polyisoprenes
)
( M , =1.6x 10'; M 2 = 1 . 4 x lo5) at 25°C. Addition of the component with
a high molecular weight raises the viscosity qr=o and produces a structural
viscosity and an inflection point in the flow curves in the region of T < lo5 Pa,
this inflection not being observed with homogeneous polymers [9]. The
numbers on the curves give the proportion (percentage) of M , .
apparently because other factors such as differences in Bueche's
slipping factor['] may play a part. The constants derived experimentally[17Jfor eq. (23) agree with the constants obtained
differently for eq. (14), in fact much too well in view of the
large limits of error in the construction of eq. (23). The size
q T = o / q T = xof the structural viscosity is thus simply equal
to the square of the number of network chains per molecule,
which is a clear indication of a cooperative process. The
experimental result does not confirm other theoretical claims
so we11[4. 51.
Angew. Chem. Int. Ed. Engl. 16, 751-766 ( l Y 7 7 )
This finding can be interpreted as showing that the same
distortion of the entanglement network is always sufficient
to reduce its effect on viscosity by a certain fraction, confirming
the considerations of the dependence of the number of active
entanglements on the extension, which lead to the assumption
that rWz1.6G,(see Fig. 4). The collapse of the entanglement
network becomes evident when, at high shear stresses, melt
fracture occurs. Consequently, these critical shear stresses are
proportional t o cp2/V;[l7 , 8 J so
as no stretch crystallization occurs.
Moreover, it can be assumed that under the influence of
an applied load, i. e. deformation of the chain fragments taking
part in entanglement, the diffusion steps or conformational
changes leading to a removal of the load are preferred to
others and so accelerate the disentanglement of the stressed
network chains[6,41,but leave the process of new entanglement
unaffected. A gradual decrease of entanglement could therefore occur during relaxation after large deformations.
'
159
99 98 95 90 -
10
180-
T
0
??So-
-.
- 8
N
5
c
=4020 -
10 -
5ZL
I
-2
-1
I
I
10
1
log
2
3
L
t [sl-
Fig. 15. Rate of strain relaxation (as a percentage) in a non-crosslinked
polyisoprene ( M / M o " 130) after momentary (5 ms) compression increases
with increasing deformation as a result of the nonlinear form of eq. (9)
and for the same reason a s causes structural viscosity (see text). i.=extension
ratio. f, = uJ;. can be calculated for i 2 0.92 by means of eq. (9) with Gr,== 3.0
bar. cm *.
The same causes as lead to the appearance of structural
viscosity lead to the fact that the relative rate of stress relaxation is greater after large than after small elongations: According to Figure 15, the halflife of mechanical relaxation decreases
with increasing
5. Influence of Entanglement on the Extension of CrossLinked Materials
The fact that the effective entanglement number depends
on the extension is also important for the properties of materials such as rubber"'] that are cross-linked by main valences.
Figure 6 shows their relaxation behavior in comparison with
that of non-cross-linked material. With slightly to medium
cross-linked samples we find, at short loading times, relaxation
moduli that correspond to those of non-cross-linked polymers.
Only after prolonged loading times does cross-linking by main
valences become noticeable, so that analytic stretching experiments are often carried out with extended loading times. On
the other hand, in chemically weakly cross-linked samples
and with brief and slight (e. g. periodic) deformations, restoring
forces are found that arise only from the entanglement network
and, according to Figure 6, are usually greater than expected
on the basis of the weak main-valence cross-linking. Since,
however, according to Fig. 4, entanglements are made ineffective by strong deformations, even with short-lasting load applications one can, by using strong deformations, recognize the
restoring forces that arise from the main-valence cross-linking.
At large extensions one then finds the extension behavior
predicted by the theory for main-valence networks" '1. With
medium extension rates and deformations one must divide
the restoring forces by the Mooney-Rivlin-Saunders procedure["] into 2 C1, the fraction due to the main-valence
network, and 2 C2, the fraction due to the entanglements
(Fig. 16).
760
I
0
I
I
I
I
I
I
,
0.5
111-
Fig. 16. As described by Mooney-Rivlin-Saunders [19], the quotients of
the stress u (referred to the cross-section of the stretched sample) and i z-(l/;.)
(;.=extensionratio)are plotted against 1,;'. The intercept 2 C , on the left-hand
ordinate is a measure of the main-valence cross-linking; that on the right,
2Cz, is a measure of the concentration of entanglements active under the
measurement conditions. 2 Cz decreases with increasing duration of stress
1121. Curves 1 to 6: d i / d t = 1 3 , 1.25, 0.12, 0.025, 6 . 2 5 ~
1.25 x
The ordinate intercept of the straight part of the curve
at i= 1 is 2C2, and that at i = 03 is 2 C1. The fraction 2C2
decreases, as expected, when the loading times are increased,
and also if the sample is swollen, as would be expected from
the strong concentration dependence of the entanglement
number shown by eq. (1 3). The experiments confirm the following
In the rubber region of relaxation behavior the breaking
stresses and maximal extensions also rise with increasing rate
of extension, because with short loading times (rapid stretching) the restoring forces affect more contacts and entanglements than with long times.
Finally, the maximum degree of swelling of cross-linked
samples is influenced by entanglement of the thread molecules
because entanglement changes the thermodynamic properties
of the solutions. In the usual swelling equations a value of
the interaction parameter (Flory-Huggins parameter) i[l
should then be
which is influenced by the effects
of entanglement, as will now be shown.
6. Significance of Entanglement for the Thermodynamic Properties of Solutions
If melts or solutions become rubber-elastic by the entanglement of thread-like molecules and thus possess an entropydetermined memory, and if the entanglement density depends
Angrw. Chem. I n t . Ed. Engl. 16,751-766 ( 1 9 7 7 )
on the concentration, then on dilution of such a network
an entropy change AS occurs that differs from the entropy
change for unentangled dissolved polymers[2o,14]. This can
have a considerable bearing on the thermodynamic properties
of such systems and will be formulated here, because previous
formulations[2' 1 of statistical thermodynamics cannot yet be
used for experimental testing.
If a chain comprising b bonds or Z segments (of bonding
number b, and length I = 1 ) is fastened at one end in such
a way that the orientation of the first segment with respect
to the anchoring is defined, then that chain can assume wo
conformations, which can be calculated when the rotation
potential has three minima and thus each bond can be oriented
in a total of three spatial
S,
= k N z Inw,,,
=k
N 2 { b,Zln3 +(m,- l)[ln0.074- lSln(Z/rn,)- l n ~ ~ , , ~ (31)
]]
At least in the case where cl,,,=l, eq. (31) agrees with
other formulations for network-formation
In addition to the entropy of mixing S,-S1 derived from
eq.(31),afurther term - k N l l n ( l --)must beintroducedinto
the equation for the free energy of mixing (subscript M), since
this describes the amount of entropy that should be assigned
to a giant molecule containing N 2 . Z segments (the entanglement network)[22]:
A G M = A H M - T A S M = k Ti N19- T [ S , - S ,
- k N , In(l-cp)]
(32)
The osmotic pressure pOs can now be calculated by substituting eq. (31) into eq. (32) and by partial differentiationlZ2]:
If such a chain is fastened at both ends, e.g. at entanglement
points, then only a fraction of the previously permissible conformations is possible. This can easily be understood by considering a chain of two segments. The end points of the second
segment mark out evenly the surface of a sphere of radius
1. On average, the first and the second segment make an
angle of 90" to each other. If we specify that this angle is
strictly retained, so that the chain ends always maintain the
statistical mean value of their separation, then the end of
the second segment can occupy only positions on an arc
and no longer on the whole surface of the sphere.
If the separation h of the ends of the chain considered
in eq. (26) has the statistical mean value, then h;=Z1'; and
with / = I and eq. (2) the conformation number we of the
chain fixed at both ends[31is given by:
(33)
Vl is the molar volume of the solvent, N ~ = 6 x
and
k is Boltzmann's constant. To carry out the differentiation
we need dsc:,,/dp [eq. (30)], ~ 2 [eq.
, (19)),
~ and finally the
quantity (a(PIaN1)N2:
(34)
We now obtain a relationship for pOsthat involves not only
the known term containing the Flory-Huggins parameter
z1[22], but also a further term that depends on the relaxation
modulus Gr,, of the solution, i. e. on its entanglement number
N, [see eq. ( 9 ) ] . f ( q ) is the experimentally determinable function defined in eq. (18).
In the case of greater entanglement each polymer molecule
with its 2 segments is divided into m + 1 = ( M / M , ) + 1 parts,
of which m - 1 parts are network chains with Z / m segments
and two parts are end fragments with Z/2m segments. The
total number of conformations of such a chain is Q,:
(35)
For calculating the dependence of wm on concentration we
require the concentration dependences of rn and x,. According
to eq. (18) we have:
The concentration dependence of d,, follows according to
eq. (19) from that of G,,,, cp, and f ( c p ) :
The entropy S , of the network of N 2 polymer molecules
and N, solvent molecules can now be obtained['41:
Angew. Chum. I n t . Ed. Engl. 16, 751-766 ( I Y 7 7 )
In the last term the part beginning with 1/(1 +...) describes
the usually small influence of the unentangled end fragments
of the network. Eq. (35) is valid only for the case of an
entanglement network and must not be extrapolated to q =0,
i. e. to a very dilute solution.
With the experimental value of Gr,q we can calculate, by
means of eq. (35) the portion of the osmotic pressure that
corresponds to the term with Gr,,,, This pressure differs little
at qn<O.l from the experimentally measured pressure (see
Tables 2 and 3), so that we must conclude that at concentrations below 10% the first term on the right in eq. (35) is
appreciably smaller than the term with Gr,cp.The parameter
must therefore have values of around 0.50, e . 8 . 0.51 for
polystyrene in toluene and 0.57 for polystyrene in decalin.
761
Table 2. O s m o t ~ cpressures of concentrated solutions at 2 5 T , given as rise heights d o f t h e solvent at 25°C [q]=viscosity
Polymer
Mw
UM
Solvent
Polystyrene
9.0.105
<:0.02
Toluene
9.0. lo5
< 0.02
Dimethylformamide
9.0.105
< 0.02
Methyl ethyl ketone
9.0.105
Decalin
< 0.02
Polybutadiene (Li)
Toluene
1.5
3
6
9
18
20
30
40
25
35
55
5
10
20
40
2
4
5
10
0.07
0.20
0.65
1.25
5.4
2.55
5.18
9.35
2.8
6.8
19.9
0.19
0.47
I SO
5.30
0.58
1.25
1.73
4.93
20.57
119
0.9 1
3.51
16.31
0.21
0.56
1.07
2.45
10.22
7.75
8.0
6.9
15.8
7.94
17.55
12.28
0.70
3.40
8.58
0.13
0.44
1.68
2.35
11.6
2.74
13.45
18
40
Polyisoprene (Li)
(IR 305)
<0.1
Toluene
0.3
Toluene
Polyisobutylene
5
10
20
4
6.7
10
I5
Polylvinyl acetate)
0.3
Toluene
Polydimethylsiloxane
0.3
Toluene
Poly(buty1 methacrylate)
0.3
Toluene
Poly(2-etbylhexyl acrylate)
Poly(dodecy1 methacrylate)
0.3
0.3
Toluene
Toluene
10.1
Cyclohexane
Polyisobutylene
0.3
Cyclohexane
Crepe [a]
0.3
Benzene
Polyisoprene (Li)
30
29.6
30
20
30
20
30
30
10
20
30
0.5
1 .o
2.0
10
20
10
20
230
119
117
98
221
750
296
101
110
74
100
320
730
580
720
[a] Polyisoprene (ci5-I ,4).
Table 3. Comparison of calculated and measured osmotic pressures
Polymer
Polystyrene
Toluene
(M,=8x106)
Decalin
Polystyrene
Toluene
( M , = l . l x 106)
Decalin
Polyisoprene
(Li, M,= 1.7 x lo6)
762
Toluene
0.25
0.12
0.06
0.028
0.01
0.25
0.12
0.06
0.028
0.01
0.25
0.12
0.06
0.25
0.12
0.06
0.50
0.20
0.10
0.05
0.01
79
24
5.5
0.6
-
80
16
3.7
0.5
-
1.5
1.9
2.2
3.1
-
1.9
2.0
2.3
2.7
..-
0.39
0.19
0.066
0.016
~
0.55
0.15
0.051
0.01 1
~
0.45
0.18
75
1.8
4.3
-_
2.2
5
4.4
0.16
76
6
-
-
-
900
88
2.0
2.8
3.3
3.5
10
0.9
-.
.-
0.61
~~
6.1
1.35
0.25
0.038
~
2.0
0.29
0.063
0.013
0.0016
1.5
0.12
0.015
0.0019
0.00012
2.0
0.29
0.063
1.5
0.12
0.015
27
2.0
0.39
0.083
0.0032
0.50
0.51
0.52
0.53
~
~
0.080
0.01 7
0.0020
0.5 1
-.
0.55
0.58
0.58
-_
-.
0.50
0.51
~
0.51
0.56
0.0124
0.0 t 3
0.00009
-
0.054
-
-
0.42
0.51
0.48
0.44
~
0.19
0.0062
Angel*'. Chem. In[. Ed. Engl. 16,751-766 ( 1 9 7 7 )
it follows that K*c/Ro is independent of M and proportional
to c, i.e. Ro is independent of both M and e[20,141 (Fig.
18). Then the diffusion coefficient D measured in concentrated
solution is also proportional to c, whereas the value obtained
by measurement in the supernatant dilute solution in an overlaying experiment is almost independent of ~ ~ ’ ~Thus,
1 . the
A theoretical f o r r n ~ l a t i o n ~ corresponding
’~~,
to the lattice
presumes that at high concentrations and with
a homogeneous segment distribution the osmotic pressure
is independent of the molecular weight and is proportional
to the square of the concentration, but with these corrected
x1 values this theory gives pressures that are much too low.
For the same reason, the excluded-volume theory cannot
explain the experimental values, especially as according to
this theory the pressures would be expected to depend on
the molecular weight. Experiment[”] shows (Fig. 17) that
above a critical concentration ‘p* the concentration dependence can no longer be described by the virial coefficients
of this theory.
I--0.001
c [g.cm-’]
Oa01
-
0.01
[g.cm-31
0.01
0.1
Fig. 18. Light scattering Ro (extrapolated to the direction of primary beam)
as a function of the concentration c for polystyrenes of various molecular
weights ( U M s 0 . 3 )in toluene at 25°C. At sufficiently high concentrations the
scattering becomes independent of the molecular weight and of the concentration [2O].
0.l
Fig. 17. Osmotic pressure p.. (represented as the rise height d of the solvent
for {)=OX?) plotted against the concentration c in toluene (polymers 1 t o
5 ) [20] and in decalin (cis-trans mixture) (polymer 5 ) at 25 “C for the following
polystyrenes: I. M,=4.1 x lo4. UM<0.02; 2. M W = 1 . 7 xlo5. Uu<0.02; 3,
M,=4.7x105,
UM<0.02; 4, M 2 = 9 x 1 0 5 , UM<0.02; 5, M,=1.4x107,
UM10.3.At sufficiently high concentrations the values reach the limiting
straight line with all molecular weights.
The term including Gr,q in eq. (35) is almost proportional
to 40’. The same is true of xl q’, so that the two terms can
be treated together by means of an effective parameter x ~ ,
The Xl,eff measured at high concentrations thus contain an
entropy component that is derived from the entanglement
network and in nonideal solutions is dependent on concentration, because there G,,$qZ is not quite constant but decreases
with increasing concentration. z ~thus
, increases.
~ ~ ~
Eq. (35) is also important for light scattering Ro (in the
direction of the primary beam) and for diffusion in concentrated solutions, as shown by the following relationship^['^^ 2 6 ]
( K * = light scattering constant):
(37)
~ ~
concentrated solution increases its volume, i. e. it swells like
a cross-linked polymer. Permeation of low-molecular substances through polymers, diffusion in melts (adhesion), and
sedimentation are also altered by the formation of an entanglement network, as can be recognized from the preceding discussions. Finally, the effects of entanglement on the demixing
behavior of concentrated solutions, e. g. on concentration of
the concentrated phase and on swelling of substances crosslinked by main valences, can also be formulated by means
of the equations given above. In an approximate manner,
~ .
an effective interaction parameter
can be used for this
purpose[14!
7. Effects of Entanglement at Medium and Low Concentrations
It now remains to demonstrate at what concentrations q**
a solution begins to form an entanglement network. Figures
9 and 19 show that an osmotic pressure independent of the
molecular weight can be observed already at small concentrations where one cannot detect the very small rubber elastic
modulus arising from an entanglement network. The limiting
concentration for the formation of an entanglement network
follows from equations (9) and (18):
Since according to eqs. (35) and (13) pOs is independent of
M and is proportional to the square of the concentration,
Angew. Chem. Inr.
Ed. Engl. 16, 751-766 ( 1 9 7 7 )
763
\
100
0
0.1-
tI
i
10
r---l
T
E
a
TI
* u 0.010-
rheology
1.c
57
0 -
.
P
*--.
690
osmosis
0.001-
44
ul
S
light scattering
no
0.1
L
/
2
1Tii
1
I
0.1
1.0
9
Fig. 19. Diagram plotted analogously to Fig. 9, showing Gr,Y7/qand pJq
forpolystyrene(Mw=8x 106,M,=5.5 x 1O6)intolueneat25"C.Theleft-hand
branch of pos/q was determined as in Fig. 9; the right-hand branch comes
from the literature 1301 (Mn=2.9x to5).
If then in accordance with eq. (13) G,,,/cp is plotted against
q, one observes that, for polystyrene with M , z 6 x lo6 in
toluene or decalin, Gr,q disappears at rp**x0.02. The critical
concentrations following from measurements of elasticity (see
Fig. 19 for G,.,+O) correspond well with those from rheology
(Fig. 10). Their value is somewhat higher than expected and
could be explained by cr,,,<l (see below), but in any case
it is far above the critical concentrations measured osmotically
or by light scattering (see Figs. 9, 19, and 20). Furthermore,
the critical concentrations measured osmotically are inversely
proportional to [q] according to Figure 20 and thus contrary
to eq. (38). The value corresponds to a state of the solution
where thecoilsjust fill the volume of the solution. For spherical
coils of diameter h this concentration would be:
q* =0.74 x
2.5
1.85
-= -
Ctl [sl
(39)
The values obtained in good solvents follow this relationship
very closely. Clearly mere mutual contact between the coils
causes disturbances that raise the osmotic pressure above
the value calculated by the excluded-volume theory (second
virial coefficients dependent on the molecular weight) and,
contrary to this theory, make the osmotic pressure independent
of the molecular weight.
The cause of the disturbances is that at the concentrations
in question the coils begin to interpenetrate; during this process
entanglements occur between the end fragments of the chains,
which may be found mainly in the outer regions of the
Such entanglements are not subject to the statistics that led
to eq. (9), so that M,, the molecular weight of the network
chains, is larger, and q* is smaller, than expected. At this
stage the entanglements need not lead to a complete network
764
1';
[I][mlil]
1
0.01
0.001
b
10
1.8
10
+
Fig. 20. Critical concentrations c* above which the osmotic pressure po3
or thelight scattering R o is independent of M ,or the viscosity of the solutions
rises with q4. plotted against the limiting viscosity [ I T ] for polystyrenes in
toluene at 25°C (A, 0 , o), in cyclohexane at 35°C (A), and in decalin (cistrans mixture) at 25°C ( x ) .
but may only bind several coils together, thereby considerably
restricting the number of possible conformations. The molecules can reduce these disturbances by shrinkage, so long
as the increase in free energy due to shrinkage does not
exceed the increase due to entanglement. Presumably some
shrinkage occurs in the neighborhood of cp*, which in the
neighborhood of cp** is largely superseded by stronger entanglement. On shrinkage, the free energy of the N 2 molecules
in the network
In expression (40) CM is a constant, X K = X , &
is a relative
expansion factor ( < 1) in the shrunken molecule and xo is
the expansion factor ( > 1 ) before the shrinkage, the latter
factor depending on interactions with the solvent. From the
assumption ai=cp*/cp that is arguable in the vicinity of cp*,
the pressure can be calculated as:
According to Table 3 this expression agrees sufficiently well
with the experimental values for polystyrene in toluene, polyisoprene in toluene, and polystyrene in decalin in the neighborhood of 'p* for zl =0.5. The a,,,< 1 discussed in respect
of eq. (38) also supports this hypothesis.
Thus for a quantitative interpretation of the experimental
pressures the relationship xK= cp*/(p must be assumed, i. e.
Angew. Chem. i n r .
E d . Engl 16. 751-766 f 1 9 7 7 )
a strong dependence of tl on concentration. As the usually
assumed decrease of tl over the whole concentration range
to x= 1 in the melt is much smaller, this concentration dependence does not suffice to explain the osmotic pressure.
It is possible that entanglements exert their effect even
at concentrations below cp* thus influencing the second virial
coefficients A 2 of the osmotic pressure. Specifically, it is
remarkable that, according to Table 2 and Figure 17 (where
it is designated by d/c), the reduced osmotic pressure pos/cp
is greater than limp,,/cp by the same percentage, indepenp-10
dently of the solvent efficiency at the same cp/cp*. In
fact, theories of the excluded volume do not take into account
the conformation restrictions due to contacts between two
chains or distant chain fragments, at least not explicitly. The
parameter
derived from A2 o r from the expansion factor
tlo (in dilute solutions) may therefore contain an entropy contribution due to conformational disturbances.
8. Abnormal Effects of Entanglement at Very High
Concentrations
At concentrations above 20 % conformational changes may
be subject to hindrance[l4I, which has not been discussed
above. At large entanglement numbers N , the distance of
the ends of the network chains of the entanglement network
is greater than the distance between spatially neighboring
network junctions, in polystyrene for example 1.4 x
cm
as opposed to 0.45 x 10-6cm (see Fig. 5). Consequently, the
m - 1 subchains of a chain cannot be arranged as freely as
is assumed in eq. (27).
It is not easy to calculate the
so it must
be estimated114]. Starting from one entanglement (network
junction) a network chain ( Z , segments) passes through several
(namely n) lattice planes of such disturbing network junctions.
The twelve nearest neighbors divide the totality of spatial
directions into g groups of directions of almost equal size.
The network chain decides in favor of one of them and thus
reduces the number of its possible conformations to the fraction l/g. It carries out this limitation n = ( M c / M * ) - 1 times,
so that the numerator in eq. (27) must be increased by the
factor 9". M* is the molecular weight of a coiled chain fragment
whose ends have the same distance as the spatially neighboring network junctions.
Reckoning that network junctions acting in this way may
include both the entanglement points N , and, to a smaller
degree (factor j < I), the n/2 quasi-network points assigned
to the n segments of the network chains, we obtain
and the distance separating the ends of the chain fragments
of molecular weight M* becomes:
and for j = 1 :
(44b)
The number n is then introduced into eq. (31), giving:
The bracket at Gr,v in eq. (35) then receives for jn/2 < 1 a
term proportional to
and for j n / 2 1~ a term that is proportional to c p [ 1 4 ] .
As a result, pOs now increases more steeply with cp than
the second term in eq. (35) would lead us to expect. In addition,
at high cp the first term in eq. (35) has a large influence
on pas, independently of the value of the parameter x,.
The n portions of the network chains which may disturb
the conformational changes of the chains need not also act
elastically, at least not at small deformations where the number
n is not altered by deformation. Figures 9 and 19 show that
at very high concentrations the experimentally determined
relaxation modulus has an almost normal dependence on
concentration, and thus it is not possible to draw any conclusions about the conformational hindrance considered here
from measurements of elasticity.
9. Final Remarks
The concept of entanglement of thread molecules is confirmed by many experimental findings and lucidly explains
some important properties of melts and concentrated solutions, i. e. of noncrystalline systems.
Chain entanglement certainly also occurs in partially crystalline polymers. This explains why polymers that have been
stretched, i. e. deformed in the crystalline state, shrink on
melting: the sample regains the form that it had before being
stretched. The entanglement network has clearly withstood
the crystallization and the deformation of the partially crystalline material and on melting appears as a deformed network
with a restoring force corresponding to its modulus G , [see
eq. (1 3)] and the deformation due to stretching.
Some effects of entanglement still remain to be treated
more precisely-e. g. those occurring in phase separation, in
intramolecularly cross-linked molecules (the effect of crosslinking so far observed for such cases is surprisingly
and in the chemical reactivity of the groups and structure
components ( e .9. double bonds) situated at entanglement
positions.
In any event, in discussions of the properties of polymers
it should always be borne in mind that entanglement can
strongly affect the properties of soft polymers and solutions.
Received: 6th May 1966 [A 182 IE]
German version: Angew. Chem. 8Y. 773 (1977)
Translated by Express Translation Service, London
For j < 1, therefore:
_ _ _ ~
[t]
4ngm.. C h n . lnt. Ed. Engl. 16. 751-766 ( 1 9 7 7 )
H . G . EILrs: Makromolekule. Hiithig & Wepf. Base1 1972.
765
[2] R. G. Kirste, W A . Kruse, J . Schelten, Makromol. Chem. 162, 299
(1972).
[3] a) A . I/: Tobolskg: Properties and Structure of Polymers. Wiley, New
York 1960; b) Revised edition by M . Hofmann, Berliner Union, Stuttgart 1967, pp. 118f.
[4] W W Grawsfey, Adv. Polym. Sci. 16, 1 (1974).
[5] a) F . Btreche: Physical Properties of Polymers. Interscience, New York
1962, p. 61; h) J. Chem. Phys. 20, 1959 (1952); 25, 599 (1956).
[6] M . Hofmann, Rheol. Acta 6. 92 (1967).
[7] P. J . Flory: Principles of Polymer Chemistry. Cornell Univ. Press,
New York 1953, p. 402.
[8] H . A . Sruurr: Die Physik der Hochpolymeren. Springer, Berlin 1953,
Vol. 2, p. 655.
[9] M . Huj>nann, Rheol. Acta 6, 377 (1967).
[lo] See ref. 171. p. 405; W Kuhn, F . Griin, Kolloid-Z. 101, 248 (1942):
cf. P. J . Flory, Angew. Chem. 87, 787 (1975).
[I 11 See ref. [3a], p. 94.
[12] M . Hofmann, Kolloid-Z. Z . Polym. 250, 197 (1972).
[I31 See ref. 171. p. 425; P . J . F l o r j , 7: G . Fox. J r . , J. Am. Chem. SOC. 73,
1904 (1951): J. Polym. Sci. 5 , 745 (1950).
1141 M . Huflmunn. H . Kriimer, R. Kuhn: Polymeranalytik. Thieme, Stuttgart
1977, Vol. 1 .
[l5] M . Hofmann, K . Rother, Makromol. Chem. 80, 95 (1964).
[I61 G. Dobrowofski, W Schnabef, Makromolekulares Colloquium, Freiburg,
March 3, 1977.
[17] M . Hofmann, Makromol. Chem. 153,99 (1972).
[18] M . Hoffmann. Rheol. Acta 6, 82 (1967).
[!9] M . Mooney. J. Appl. Phys. 1 1 , 582 (1940); R . S. R i d i n , D . $4Saunders,
Philos. Trans. R. SOC.London A243.251 (1951).
[20] M . Hqffmann, Makromol. Chem. 174. 167 (1973).
[21] a) A. S. Lodge, Rheol. Acta 7 , 379 (1968); b) R. Takserman-Krozer,
A . Ziabicky, J. Polym. Sci. A2, 7, 2005 (1969); A2, 8, 321 (1970); c)
F. S. Edwards, Proc. Phys. SOC.London 85, 613 (1965); 91, 513 (1967);
92, 9 (1967); Discuss. Faraday Soc. 49, 43 (1970); J. Phys. A 6 , 1169,
1186 (1973).
1221 See ref. 171, pp. 577, 512; P. J . Flory, J . Chem. Phys. 18, 108 (1950).
1231 M . L . Huggins, J. Phys. Chem. 46, 151 (1942).
[24] M . Dauuil, J . P Cotton, €3. Farnoux, G . Janninck, G . Sarmu, H . Benoit,
R. Dupfessiu, C . Picot, P. de Gennes, Macromolecules 8, 804 (1975).
[25] P. Debye, J. Phys. Colloid Chem. 51, 18 (1947).
1261 G. K Sehufz, 2. Phys. Chem. 193, 168 (1944).
1271 See ref. [5a], p. 19; P. Debye, F. Bueche, J. Chem. Phys. 11, 470
(1943); A. Ishihora. J. Phys. SOC.Jpn. 5, 201 (1950).
1281 See ref. [7], p. 599.
1291 G. Gee, L . R. G. Peloar, Trans. Faraday SOC.38, 147 (1942); G. Gee,
W I . C . Orr, ibid. 42, 507 (1946).
1301 C . E . H . Bawn, R . F . J . Freeman, A . R . Kamallidin, Trans. Faraday
SOC.46, 677 (1950).
1311 J . Furukawu, H . Inaguki, Kautsch. Gummi Kunstst. 29.744 (1976).
[32] U . Eisele, Lecture at the IISRP Conference Williamsburg (USA) 1976.
Flexible Drug Molecules and Dynamic Receptors[**]
By R. J. P. Williams[*]
When a small flexible drug molecule binds to its likewise mobile receptor (protein, membrane
etc.) the shape and function of both can change. The study of the nature and extent of these changes
by several independent methods gives an insight into the mode of action of drugs. The static
lock and key model will most probably have to be revised or be replaced by the nebulous
concept of dynamic states.
1. Introduction
The basic problem of drug action in biological systems
is easily formulated. In order to interact with a biological
system any drug must first bind and may then react, whence
a comprehensive minimal two-stage reaction path can be
written
where D is the drug which undergoes a rapid reversible binding
to a biological receptor L, i.e. a protein, DNA, RNA or
a membrane etc., giving DL. This reaction has an equilibrium
binding constant
[*] Prof. Dr. R. J . P. Williams
Inorganic Chemistry Laboratory
South Parks Road, Oxford OX1 3QR (England)
[**] This article is based on the Merck, Sharp and Dohme Scientific Lecture
1976 given in London. It was also given in outline at the Belgian Chemical
Society Meeting in Namur, 1976.
766
and there can be many successive steps of this kind before
the 'final' binding condition is reached. The second step in
the above simplified scheme [eq. (I)], which may or may
not be a required part of action, is an irreversible chemical
combination of D in the form D' with a part of the biological
system L', where L' may be a protein, RNA, DNA, a membrane etc. which has incorporated D (or D'). L and L' could
be the same receptor site of course. The reaction rate constant
in its simplest form is a first order rate constant, kDL, but
again many such steps could be involved.
My analysis of drug action will be based upon the structural
features of these reaction paths starting from the structures
of D and L themselves. By structural features I imply the
whole series of conformational states through which the two
species D and L must go in order to reach some final state
DL or D'L'. Recently the nature of such pathways has
been stressed by Feeney, Roberts and Burgen"', who were
motivated by their observations, using nuclear magnetic
resonance (NMR) spectroscopy, on the mobility of certain
drug and hormone molecules. My independent and parallel
interest in the problem of conformational mobility has arisen
through studies of both small and large molecules in solution
also using sophisticated NMR methods for conformational
analysis. In the past the solution structures of D and L (which
Angrw. Chem. I n t . Ed. Engl. 16, 766-777 i1977 J
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