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heterogeneities
Akira Onuki
To cite this version:
Akira Onuki. Scattering from deformed swollen gels with heterogeneities. Journal de Physique II, EDP Sciences, 1992, 2 (1), pp.45-61. �10.1051/jp2:1992112�. �jpa-00247611�
Classification Physics Abstracts
03.40 05.20 82.70G
Scatterblg from deformed swollen gels with heterogeneities
Akira Onuki
Department of Physics, Kyoto University, Kyoto 606, Japan
(Received is July1991, accepted 9 October1991)
Abstract. When elastic materials are anisotropically deformed, heterogeneities in stiffness
generally produce long-range density variations. The effect is particularly enhanced in swollen
gels. We calculate the structure factor from such frozen density deviations under general alline deformations. It is maximized in the most stretched direction, whereas that from the thermal fluctuations is minimized in the same direction. As a result, under uniaxial extension, a crossover
occurs from normal to abnormal butterfly scattering patterns with increase of the strength of
inhomogeneity or the swelling ratio in accord with recent scattering experiments. Further-
more, scattering patterns under shear deformation closely resemble those from phase-separating
semidilute polymer solutions under shear flow.
1 Introduction.
Recently small-angle neutron scattering experiments have been performed on uniaxially
stretched network systems such as crosslinked polymer blends or swollen polymer gels in a
good solvent [i-3]. In some cases an abnormal increase of the scattering intensity I(q) has been found in the stretched direction at small
wave numbers q. Here we take the z and y
axes parallel and perpendicular to the stretched direction. In such cases scattering isointensity
curves on the qx qy plane have unusual shapes known as "butterfly patterns". They are
characterized by Ijj(q) > Ii (q) at small q and Ijj(q) < Ii (q) at large q, where Ijj(q) and Ii (q)
are the intensities ivitl~ q along the x and y axes, respectively. It should be noted that, if
homogeneous gels with a constant crosslink density are deformed, the thermal density fluctu- ations are suppressed in the stretched direction, leading to Ijj(q) < Ii (q) at any q [4, 5]. The resultant scattering pattern may be called the normal butterfly pattern, whose direction of
anisotropy is opposite to that of the pattern observed in the experiments [5]. Such anisotropy
at small q is enhanced on approaching to a bulk instability point of the volume phase transition [6]. On the other hand, Bastide and Leibler iii pointed out that small-scale heterogeneities in
the crosslink density of unswollen gels result in large-scale density variations when the gels are expanded. Afterwards, Bastide et al. [8] argued that, when swollen gels are furthermore uniax-
ially stretched, these density variations have an extended correlation length fjj in the stretched
JOURNAL DE PUYBtQUE t> T 2, N' ', JANV'ER t992
direction longer than that fi in the perpendicular directions. They concluded that these
anisotropic fluctuations should give rise to the abnormal butterfly pattern. The fluctuations treated by Bastide et al. are those frozen aroupd heterogeneities and should be discriminated from the thermally excited fluctuations. Static scattering detects the sum of these two kinds of fluctuations. However, dynamic scattering will detect only the latter thermal fluctuations
which undergo a diffusive relaxation [9]. Regarding this aspect we mention a dynamic light scattering experiment on stretched gels by Takebe et al. [10], who found suppression of the
thermal fluctuations in the stretched direction in accord with references [4, 5].
The aim of this paper is to calculate the frozen density deviations around heterogeneities
and show that they indeed give rise to the abnormal butterfly pattern. Hence this paper serves to supplement the ingenious but very intuitive theory by Bastide et al. From the mathematical
difficulty, however, we will assume "weak inhomogeneity" and use a perturbation theory with respect to the strength of inhomogeneity.
2. Elastic body With inhomogeneous elastic moduli.
Before treating a model of gels, we start with a simple, instructive example in the usual linear elastic theory assuming only small strains. This is appropriate because of rather heavy mathe- matics of nonlinear elasticity in gels. Let an isotropic elastic body have weakly inhomogeneous
bulk modulus It (r) and shear modulus p(r). The elastic stress tensor «;; is of the form,
«;; = 1<g6;; + p[v;u; + v;v; (g6;;]. (1)
where V; = 0/0z; and
g = V u
=
~j V;u;. (2)
j
The displacement vector u is assumed to vanish in some isotropic reference state for simplic- ity. We then apply an anisotropic external stress, which gives rise to an affine deformation
characterized by a constant strain tensor A;;,
(£u;I
= A,>, (3)
>
where < > is the spatial average. In deformed states small-scale inhomogeneities in It and p should induce random strains, which are determined by the mechanical equilibrium condition V. ii= 0
or
(It + j~)V;g + ~v~ui + gV;K + £(V;u; + V,u; )gb;;)(V;~)
= 0. (4)
j
The last two terms have arisen from the spatial dependence of It and ~ and will be crucial in
the following.
We assume that the deviations bit
= K It and 6~ ~ fi are small, where It and fi are
the spatial averages of the moduli. To first orders in 6K and 6p, (4) may be rewritten as
(k + fi)Vjg + fiv~u; ££ -jV;K £(A;; + A;; )6;;j)(V;p), (5)
~
j
(6)
where
g =< g >= £ Ajj
j
We take the divergence of (5), perform the Fourier transformation, and obtain
gq = -(K + fi)~~i§Kq + X(I)Pqj, ~~~
where gq, Kq and pq are the Fourier transforms ofbg = g j, SK, and bp. The function x(() depends on the direction § = q~~q of the wave vector q as
x(4) = L(A;> + A>I )b;>#)fl. (8)
For eXample, around a single heterogeneity localized at the origin r = o,gq depends only on (
at long wavelengths. This means that the inverse Fourier transform 6g(r) is a function propor- tional to r~~ and dependent on the angle-coordinates far from the origin. Some calculations
yield
6g(r) E+ (»1/8«)(R + ~j#)-~L(Au + A>1 )6i>#)z;z>j (9)
I,j where we have set 6p(r) = p16(r).
Because the density deviation bpis given by -pbg to first order in the deviations, we readily obtain the structure factor If(q) =<( bpq (~> for the frozen density fluctuations in the form,
If(q) = P~(It + ~#)~~ [#~GKK(q) + 2gx(4)GK~(q) + x(4)~G~~ (q)]. (lo)
Here we define the correlation functions of the modulus deviations,
GKK(q) =<I I<q l~>, GK~(q) =< I<qPi >, G~~(q) =<I Pq l~> (ii) They have unique small-wavenumber limits if the modulus deviations have only short-ranged
correlations. Nevertheless, If(q) still depends on ( through x(().
In a uniaxial state represented by A~~ = sjj,Ayy = Azz = c i, and A;; = 0 for I # j, we find
§ = cjj + 2ci and
X(4) " 2(cjj si)(cos~b ~), (12)
3 where cosb
= qx/q. The absolute value of x(() takes a maximum at b
= 0 in the region
0 < b < 1r/2, so the intensity can take a maximum at = 0 in wide parameter regions.
Namely, the intensity can indeed be increased in the stretched direction. This tendency will also hold in swollen gels in some parameter regions. From (9) the density deviation around a localized heterogeneity at the origin is proportional to (2z~ y~ z~)/r~ far from the origin.
The absolute value of 6g(r) is maximized in the stretched direction. On the other hand,
under a shear deformation represented by u~ = 7y and uy = uz = 0, we have j = 0 and
X(4) " 27qxqy/q~. The intensity is maximized for q~/qy = +I and qz = 0 irrespectively of the
sign of 7.
Without heterogeneities the structure factor from the thermal fluctuations Ith(q)
assumes kBT/(It + ~p) as q
- 0. If I( and ~ are weakly inhomogeneous, it may be writ- ten as Ith(q) " kBT/[(I( + )p £(q)], where £(q) in the long wavelength limit is
E(q) £t (k + (#)~~l< bI<(r)~ > +(
< b»(r)~ >l + ~
< b»(r)~ > +O(q~). (13)
Here the correlation functions in (ii) have been assumed to be independent of(. The averages
< > on the right hand side of (13) are taken over the space coordinates and they are
constants independent of r. See Appendix A for more details.
3. Polymer gels With inhomogeneous crosslink density.
3. I FROZEN DENSITY VARIATIONS. We expect that the frozen fluctuations can be much
more important in swollen gels~ where large deformations can be easily produced. To use a classical rubber theory ill, 12] we first neglect the excluded volume effect supposing theta or poor solvents. However, scaling considerations will allow us to qualitatively extend our results to the case of good solvents [13, 14]. Hereafter, xo " (z(, xi,~() are the Cartesian coordinates in some reference relaxed state representing the original position before deformation, while X = (Xi,X2,X3) are those of the deformed gel representing the real spatial position. The
polymer volume fractions in the two states are written as lo and #, respectively. The simplest
form of the elastic free energy is given by ill, 12, 15],
Fei = lkBT £~dxov(xo>
Ii
I]x>)~ in«of<)] <14>
where the integration region is within the volume Vo of the relaxed state and v<xo> is the crosslink density dependent on xo. We shall see that its inhomogeneity produces large elastic strains in swollen states. Flory introduced the logarithmic term (c~ in <#o/4» in <14) to
account for an entropy of springs [12]. In our case it is required in order to make the elastic
stress, the second term of (17) below, vanish in the relaxed state. The total free energy is then
F = Fet + kBT / dX f<#)
+ fnh, <15)
v
where the integral in the second term is within the volume V of the deformed gel, kBTf<#)
is the mixing fi.ee energy density dependent of # <and the temperature) [6], and F,nh is the
gradient free energy. The # is'related to the determinant of the matrix @X~lox) as
loll " Det(@Xil@x)). <16>
We first neglect F,nh and calculate strains only in the long wavelength limit, q - 0. The stress
tensor a;; due to deformations is then of the form [15],
a>; = if 6;> + v(xo>) l~((xi) []x;) >jj
<17>
~ ~ ~
Against an infinitesimal displacement 6X, the change of F should be written as
bF = / dX~ja;j ((6X~),
<18)
v ; ; ;
where bX is regarded as a function of X. From this relation la;;) may be calculated as <17) if fnh is neglected. As noted above, the second term of <17) vanishes in the relaxed state owing
to the logarithmic term in <14).
We deform the gel allinely on the average as
l~qX;) = ru> (19)
,
where < > is the spatial average. The determinant of (r;;) is the volume expansion ratio,
so we define the linear expansion ratio I by l~
= Det(r;;) =< loll > <20)
The I can be much larger than I after swelling. The mechanical equilibrium condition V. $= 0
is written as [lS]
~
~
l 0
,~j~~)j) ~j £"'~°~/~~' ~' ~~~~
j~f"q4 j@
j
where f" = @~f/@#~, and # and VI are regarded as functions of X in the first two terms.
This condition is equivalent to the extremum condition 6F/bX = 0, where F is regarded as
a functional of X(xo) [15]. We use x
= (xi, x2, x3) instead of xo to denote the average space
position,
x; = ~jT;;x). (22)
I
We consider the displacement u = X x only to first order in the deviation bv
= v fi of the
crosslink density, fi being the average. Note that
g + v u = z )u;
s+ -6</<. 123)
,
Hereafter the average volume fraction is simply written as #. Using (23) we may transform
(21) into
[(loll" + fi)V~ + RV()g l£ ~j<b;; Ei;)<ViV; v), <24)
;,;
where V; = 0/0x; and
V( = ~j<@/@x))~ = ~j E;;ViV;. <25)
I ",j
The matrix E;; is written in terms of r;; defined by (19) as
E;; = ~j r;m r;m, (26)
m
which is a synunetric, positive-definite matrix. The Fourier transform gq of g(x) is thus given by
gq + 1404f" + R + R'~J14)l~~ Ii '~J(4)lvq, (27)
where I = (#o/4)~/~>vq is the Fourier transform of 6v, and
J(« =
j ~ E;;
q;q;. (28)
,,
By an orthogonal transformation (U;; ), the matrix (E;; may be diagonalized. The three eigen values are written as l~Ji, l~J2, l~J3 with 0 < Ji < J2 < J3. From Det(E;j
= [Det(T,,)]~ it follows that
JiJ2J3 = 1. <29)
By the transformation, q( = £j U;jqj, we should have
J14) = ~ J;lql/q)~ 13°)
Thus the maximum of J ii) is J3 in the most stretched direction and the minimum is Ji in the most shrunken direction.
If the gel were homogeneous and were isotropically expanded, it would have the following
bulk and shear moduli [lS], k
= k~T[12 /" + Pi /io )Pi-11, j31>
p = kBTRI~~ <32>
We may derive these relations from (17) by considering small deformations around an isotropi- cally swollen state. We define k and p by these expressions even in deformed, inhomogeneous
states and rewrite <27> as
9q " It + jfi + fiJ14)1~~ lfilR>l~~~ J14>1"q. <33)
This relation is analogous to (7> if we notice J<(> £t I + x<4) for slightly anisotropic deforma- tions and vq/fi = pq/p for fixed 1.
3.2 STRUCTURE FACTOR IN THE LONG WAVELENGTH LIMIT. We hereafter assume that
the correlation of bv is of short-ranged in the relaxed state as
< 6v(xoo)6v(xo + xoo) >= 6(xo)tip. (34)
The dimensionless number p represents the degree of irregularity of the crosslink structure.
Crudely speaking, tip is the defect density in the relaxed state and our perturbation scheme will be valid only when p « I. See Appendix B for more discussions on these points. Using
the relation 6(xo) " 6(x)#o/4 from (16) and (22), we obtain
<I vq l~>E+140/4)°P. <35)
Then, the structure factor If<q) =<( #q (~> for the frozen density fluctuations is written as
Iflq) E+ 4~vi~PiE + J<I)i~~iJli) i~~i~, <36)
where
us = 14/40)°
= R/i~, 137)
E = 4~f"14)'/°
=
) + ). 138)
The us is the average crosslink density in the swollen state.
On the other hand, in homogeneous gels the thermal fluctuations of # give rise to the
following structure factor at small q,
it~<q) Et <~/i<~f" + fi>-iJii)i
= 1<~/vs>~)/i~ + Jii)1 139) We shall see in Appendix C that the thermal fluctuations are not much affected by the het-
erogeneities as long as p « 1. Then the total intensity I(q) is the sum If(q) + Ith(q), so
that
Iiq) E£ (<~/v~>~)~
+
~q~
+ P. ~ltj~l~ l 140)
where
P" " Pl~
" P(40/#)~/~ (41)
It will be important that, if I » I, p° can be of order I even for p « I. We remark that the thermodynamic stability requires E + J(() > 0 for any directions, so that
e + Ji ? 0. (42)
Before examining anisotropic cases we compare the two contributions in the
isotropic case, J(4)
= 1, as
in~(i~/i~~) = p(>2 + >-2 2)/(~ + i). (43)
This relation is in fact consistent with figure 3 of reference 3, which shows that If
= 0 in the
unswollen case I
= I and that the above ratio is about I at l~ E£ 2.
As a simplifying result the angle-dependence of I(q) arises only through J
= J((), so we
may examine general behavior of the intensity as a function of J. In Appendix D we shall find three characteristic cases in the behavior of I(q). (I) The normal case. The I(§) is a
monotonously decreasing function of J in the interval Ji < J < J3 at fixed I,E, and p° when either of the following two conditions is satisfied,
E + l~~ j 0, (44)
P° $ P3 + )lE + J3)/ile + i~~)lJ3 i~~)I 145)
The first condition ~vill be realized near the spinodal point of the volume phase transition [6, 15, 16]. The positivity of E + l~~ is assumed in (45). We also note J3 > l~~ from J3 > and I > I. (ii) The ab?iormal case. On the other hand, I(() is a monotonously increasing function of J in the interval Ji < J < J3 when E + l~~ > 0, Ji > l~~, and
P" ? Pi + j(E + Ji)/llE + '~~)lJi '~~)l. (46)
Note that (46) can be satisfied even for small p
= p°/l~ if e 2 and I » I (or at relatively high temperatures and for large expansion). (iii) The intermediate case. In the remaining
parameter regions I(q) attains a minimum at J
= Jm % il(2fp° -1) where f % e + l~~ and Ji < Jm < J3. The intensities at J = Ji and J3 coincide particularly for p* = pc, where
Pc = jPiP3/(Pi + p~j
= <E + Ji)<E + J3)/(E + l~~)[(E l~~)(Ji + J3) + 2JiJ3 2el~~]. <47)
P~
4
(II) Abnormal 2
(Ij
~~m~j (III) Intermediate
, ,
~ 2
'
Fig. 1. The three regimes are shown in the e p° plane at l~~
= o-I, Ji l~~
= 0.5, and
J3 l~~
= 4. In the normal region <I) the intensity in the long wavelength limit decreases with
increasing the degree of stretching as a function of the angle 9
= cos~~<q~/q). In the abnormal region
<II) the opposite holds. In the intermediate region <III) a minimum is attained at an intermediate angle. We have I<Jmm) < I<Jmax) above the middle curve inside <III) and the opposite below it.
If p° > pc, we have the abnormal relation I(Ji) < 1(J3). In figiJre I we display these three
regions in the E p° plane for l~~
= 0.I, Ji l~~
= 0.5, and J3 l~~
= 4. The middle curve is p* = pc determined by (47). The figure shows that, if E 2 1, a crossover occurs from the
normal'to abnormal regions with increasing p* = pl~ and that the normal behavior is expected for E < 0 <or near the bulk instability point) even for p° 2 1.
When Ji < l~~, the gel is more compressed in the least stretched direction than in the relaxed state. In this case I<() is always minimized at an intermediate J as long as p* is above
the curve p* = p3 land there is
no abnormal region in the above definition). The relation
I(Ji) < 1(J3) still holds for p* > pc with E > [l~~<Ji + J3) 2JiJ3]/<Ji + J3 21~~), where pc is defined by <47>.
Finally let us consider a large-deformation case given by Ji « E « J3 and E » l~~.
Here, I(Ji) ££ <4~/vsl~)/E in the least stretched direction and 1(J3> E£ (4~/v~l~)(1/J3 + P*>
in the most stretched direction. Therefore,
IlJ3)/IlJi) E+ El( + P°). 148)
The above ratio exceeds I for p* > I/E 1/J3 ££ 1/E surely in accord with <47).
3.3 GENERALIZATIONS TO HIGH WAVE NUMBERS AND TO GOOD SOLVENTS. Since the
gradient free energy F,nh in (15) has been omitted, our expression so far are valid only in the limit q
- 0. However, ifq is larger than the inverse correlation length, fnh is dominant in the free energy and we expect gq c~ vq/q~ instead of (27). Although not well justified, the overall
behavior of Iii) may be described by
1(~) = i<~/vs>~) ~
+
iq~~+
cj«~~ + P*(~ +ill)+ii«~~) l 149)
