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Dynamic structure factor of crosslinked polymer blends
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Physica A 373 (2007) 153–164 www.elsevier.com/locate/physa
Dynamic structure factor of crosslinked polymer blends M. Benhamou, M. Chahid Laboratoire de Physique des Polyme`res et Phe´nome`nes Critiques, Faculte´ des Sciences Ben M’sik, B.P. 7955, Casablanca, Morocco Received 2 April 2006 Available online 5 June 2006
Abstract In this paper, we are interested in the critical behavior of the dynamic structure factor of a crosslinked polymer blend made of two chemically incompatible polymer A and B, when it is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. Since the critical fluctuations occur over distances smaller than the mesh size (microdomains size), x , the dynamic structure factor should be governed by a short-time behavior we want to determine. We demonstrate that the final time, t , necessary to the appearance of microdomains alternatively rich in A and B-polymers, scales as t xz , with z a dynamic critical exponent. The investigation of the dynamic structure factor is first achieved using a mean-field approach, based on an extended Van Hove theory, and second by a scaling argument. The Van Hove theory is valid as long as the fluctuations of composition can be underestimated. Within the framework of this model, we determine an exact form for the dynamic structure factor, and in particular, we find that the corresponding dynamic exponent is z0 ¼ 6. Second, the study is extended to the case of crosslinked polymer blends of low-molecularweight, where fluctuations of composition are strong enough near the spinodal temperature. Using a scaling argument, we prove that the scaling law for the dynamic structure factor is Sðq; tÞ ¼ qZ2 f ðqRðtÞÞ, with Z the standard Ising critical exponent, f ðxÞ a universal scaling function, and RðtÞt1=z some time-characteristic length. The latter can be interpreted as the size of instabilities domains at time t , and it becomes of the order of x at time t . The growing process of instabilities is then stopped at the final time t . We show that the dynamic exponent z is not trivial and has as three-dimensional value z ’ 5:969 0:001. In dimension 2, we find an exact value for this exponent that is z ¼ 23 4. r 2006 Elsevier B.V. All rights reserved. Keywords: Crosslinked polymer blends; Microphase separation; Dynamic structure factor
1. Introduction The physical system we consider is a mixture made of two polymers A and B of different chemical nature, crosslinked in their one-phase region by controlled irradiation. For the sake of simplicity, we assume that chains A are crosslinked to chains B, and there are no crosslinks between chains of the same kind. Thus, complications related to an eventual existence of crosslinks between chemically identical chains are not considered here. When the temperature of the system is lowered, the mixture has tendency to phase separate. Because of the presence of crosslinks, the elasticity of the network resists to such a separation. As result, the Corresponding author. Tel.: +212 22 704671; fax: +212 22 704675.
E-mail address: [email protected] (M. Benhamou). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.05.015
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crosslinked mixture undergoes a microphase separation (MPS), which is characterized by the appearance of microdomains alternatively rich in polymers A and B. The pioneered theory for studying the MPS was due to de Gennes [1], followed by several extended works [2–14]. The central idea of the author was an analogy between the crosslinked chains and a dielectric medium. The theoretical predictions was tested by Briber and Bauer [15] by small-angle neutron-scattering experiment on the PS-PVME mixture. The de Gennes’ theory agrees with experiment for all wave-vectors range except in the zero-scattering-angle limit. More precisely, the theoretical structure factor vanishes at this limit, while the experimental one does not. The discrepancy between theory and experiment was solved in a series of published papers [4,6,7,9]. Most works have been addressed to the static aspect of MPS. This means that only few studies were dedicated to the investigation of kinetics of such a transition [3,5,12,14], when the temperature is changed from an initial value T i towards a final one T f very close to the spinodal point. As noted in a previous work [3], the reptation motion of a cross-linked chain is frozen by the presence of permanent crosslinks. Hence, the only motion allowed to the cross-linked chains is of Rouse type. Kinetics was studied through the relaxation rate, as a function of the wave-vector. In this paper, we reexamine kinetics problem of MPS, in term of dynamic structure factor, when the system is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. The fact that the critical fluctuations occur over distances smaller than the mesh size (microdomains size), x , we expect that the dynamic structure factor is governed by a short-time behavior we want to determine. As we shall show below, the final time, t , necessary to the appearance of microdomains alternatively rich in A and B-polymers, scales as t xz , with z a dynamic critical exponent. This time-scale has a different meaning, when it is compared to the standard relaxation time concerning those critical systems exhibiting a phase transition occurring at vanishing wave-vector. The main difference comes from the fact that the latter diverges at the critical point [16,17], but t does not. Indeed, the relaxation time obeys a behavior similar to the above one (with another dynamic exponent), where x is replaced by the thermal correlation length xt . We shall demonstrate below, at the spinodal temperature and at scales smaller than the mesh size x , the leading critical behavior of the dynamic structure factor, Sðq; tÞ, depends only on two kinds of lengths, namely the wavelength q1 and RðtÞt1=z . The latter can be interpreted as the size of instabilities domains at time t, and it becomes of the order of x at time t . The growing process of instabilities is then stopped at the final time t . The investigation of dynamic structure factor will be first apprehended using a mean-field approach, based on an extended Van Hove theory [18]. Within the framework of this model, we compute exactly the dynamic structure factor, and in particular, we find that the corresponding dynamic exponent is z0 ¼ 6. Second, we shall extend the study to the case of an eventual presence of strong fluctuations of composition near the spinodal temperature. This will be done using a scaling argument. In particular, we find that the dynamic exponent z has as three-dimensional value z ’ 5:969 0:001. In dimension 2, we show that this exponent is exact and has the value z ¼ 23 4. The remainder of presentation proceeds as follows. In Section 2, we recall the essential of the static study of MPS within the framework of de Gennes theory. The exact computation of the dynamic structure factor, using mean-field theory, is the aim of Section 3. We examine, in Section 4, the dynamical scaling. Some concluding remarks are drawn in the last section.
2. De Gennes theory Consider a mixture of two chemically incompatible polymers A and B, cross-linked in the one-phase region. When the system is cooled down, below some critical temperature, a MPS takes place. Such a transition results from a competition between the usual phase separation and the elastic properties of the gel. Following de Gennes [1], the free energy functional of the system, in a Landau approximation, writes Z F0 ½j 1 ¼ (1) dd rft0 j2 þ ðrjÞ2 þ C 0 P2 g, kB T 2
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where T is the absolute temperature, kB the Boltzmann’s constant, and r 2 Rd the d-dimensional positionvector of the representative point of the system. The notation t0 ¼ a2 12ðwc wÞ
(2)
means the temperature parameter, where a is the monomer size, w the standard Flory interaction parameter, which is inversely proportional to the absolute temperature T, and wc ¼ 2=N its critical value if the system is uncrosslinked (N being the common polymerization degree of chains). The field or order parameter jðrÞ is directly proportional to the local fluctuation of composition rðrÞ ¼ FðrÞ Fc , that is jðrÞ ¼ a1d=2 rðrÞ. Here, F is the composition of one component of the mixture and Fc ¼ 12 its critical value. We recall that the first two terms in relation (1) constitute the expansion of the usual Flory–Huggins free energy [19,20] around the critical composition, and the third one describes the gel elasticity. The elastic contribution in this relation was first introduced by de Gennes by analogy with the polarization of a dielectric medium. The internal rigidity constant C 0 was related to the average number n of monomers per strand by Ref. [1] C 0 a4 n2 .
(3)
By strand, we mean the section of a chain between two consecutive crosslinks. The quantity 1=n was interpreted as the crosslinking density [1]. In equality (1), the vector P denotes the average displacement between the centres of masses of A and B strands; it is the analog of the polarization in the dielectric problem. The monomers A and B play the role of charges ðþÞ and ðÞ of the dielectric medium, respectively. The charge fluctuations correspond to the fluctuations of composition. The quantities P and r are not independent each other, but related by the Maxwell’s relation: div P ¼ rðrÞ. In reciprocal space, the latter writes: iq Pq ¼ rq , where rq ¼ a1þd=2 jq is the Fourier transform of the local fluctuation of composition rðrÞ ¼ a1þd=2 jðrÞ and q the wave-vector. In reciprocal space, the free energy (1) becomes Z F0 ½jq 1 2 dd qS1 ¼ (4) 0 ðqÞjjq j 2 kB T with S0 ðqÞ ¼
t0 þ
q2
1 þ C 0 q2
(5)
the structure factor. This differs from the usual one ðC 0 ¼ 0Þ by the extra term C 0 q2 taking into account the permanent presence of crosslinks. It exhibits a maximum at the finite value [1] 1=4
q0 ¼ C 0 , x0
(6) q1 0
pffiffiffi 1=4 C 0 a n,
x0
whose inverse, ¼ ¼ measures the size of microdomains A or B. The length is of the order of the size of strands (mesh size), that is x0 ¼ an1=2 . The structure factor S0 ðqÞ at q ¼ q0 diverges at the spinodal for Ref. [1] pffiffiffiffiffiffi t0s ¼ 2 C 0 . (7) The divergence of the structure factor at nonzero value of q is interpreted as a MPS [1]. The above spinodal equation tells us that the crosslinked mixture is more miscible than the uncrosslinked one. Finally, the structure factor can be rewritten on the following useful form S0 ðqÞ ¼
1 , 2 2 q2 ðq2 q2 0 Þ þ x0
(8)
with x0 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi t0 t0s
(9)
the thermal correlation length. Formula (8) suggests that, in the critical domain around t0 ¼ t0s and q ¼ q0 , the static structure factor depends on three kinds of parameters, which are the wavelength q1 , the microdomains
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size x0 and the thermal correlation length x0 . On the other hand, relation (8) can be written on the scaling form S0 ðqÞ ¼ qZ0 2 Xðqx0 ; qx0 Þ;
Z0 ¼ 0,
(8a)
with the universal two-factor scaling function Xðx; yÞ ¼
1 . y4 ðy2 1Þ2 þ x2
(8b)
In particular, at criticality and at scales smaller than the mesh size x0 , the static structure factor behaves as S0 ðqÞqZ0 2 ðZ0 ¼ 0Þ;
ðq1 5x0 ; t0 ! t0s Þ.
(8c)
The next step consists in the computation of the dynamic structure factor of a critical crosslinked polymer blend, using an extended Van Hove theory [18]. The latter is valid as long as we are concerned with highmolecular-weight polymer chains. 3. Extended Van Hove theory We start with a crosslinked polymer blend out of equilibrium. Denote by jðr; tÞ the local composition fluctuation of crosslinked A-chains. These, for example, diffuse with a local current J. The latter is related to jðr; tÞ by the standard continuity equation qjðr; tÞ þ r Jðr; tÞ ¼ 0. qt
(10)
Here, r is the position vector of the representative point of the medium and t the time. On the other hand, we recall the relationship between the current J and the chemical potential mðrÞ ¼ dF0 =djðrÞ (d=dj means the functional derivative), where F0 is the free energy (1), Z 1 dd r0 Lðr r0 Þrmðr0 Þ. (11) JðrÞ ¼ kB T Here, L means the Onsager transport coefficient to be discussed below. Now, combining relations (10) and (11) yields the Langevin equation Z qjðr; tÞ dF0 ¼ dd r0 Dr0 Lðr r0 Þ þ nðr; tÞ, qt djðr0 Þ
(12)
where D is the Laplacian operator. We have introduced a stochastic force (or noise) nðr; tÞ. It will be convenient to examine the problem in reciprocal space. Fourier transforming the above differential equation yields qjq ¼ q2 LðqÞðt0 þ q2 þ C 0 q2 Þjq þ nq ðtÞ. qt
(13)
This constitutes an extension of the well-known Van Hove equation (model C) [18]. The stochastic force is assumed to be Gaussian, that is hnq ðtÞnq0 ðt0 Þi ¼ 2q2 LðqÞdd ðq q0 Þdðt t0 Þ.
(14)
Equalities (13) and (14) show that physics depends on the knowledge of the Onsager transport coefficient LðqÞ. To specify its form, we first note that, due to the permanent reticulations of chains, only local motions are possible. Therefore, the Rouse motion of connected chains dominates, and then, the diffusion is essentially governed by free monomers. At small-scales, that is for q1 5x0 , LðqÞ behaves as LðqÞGq2 [3], where G is the kinetic coefficient. With these considerations, the basic equation becomes qjq ¼ Gqb ðt0 þ q2 þ C 0 q2 Þjq þ nq ðtÞ, qt
(15)
hnq ðtÞnq0 ðt0 Þi ¼ 2Gqb dd ðq q0 Þdðt t0 Þ
(16)
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with the exponent b ¼ 4. Notice that the model used here is slightly different from the usual ones describing nonconserved ðb ¼ 0Þ and conserved order parameters ðb ¼ 2Þ. We are interested in the dynamic structure factor defined by S0 ðr r0 ; t t0 Þ ¼ hjðr; tÞjðr0 ; t0 Þin .
(17)
We denote by S0 ðq; oÞ ¼ hjðq; oÞjðq; oÞin
(18)
its Fourier transform. The notation hin means the average over the stochastic force. Then, in the total reciprocal space, the solution to the motion equation (15) reads nq;o . (19) jq;o ¼ io þ Gqb ðt0 þ q2 þ C 0 q2 Þ The use of Eq. (16) leads to the dynamic structure factor S0 ðq; oÞ ¼
2Gqb o2 þ ðo0c Þ2
(20)
with b 2 2 o0c ¼ Gqb S1 0 ðqÞ ¼ Gq ðt0 þ q þ C 0 q Þ
(21)
the characteristic frequency, which defines the half-width at half-height. This frequency may also be rewritten on the scaling form o0c ¼ Gqz0 Oðqx0 ; qx0 Þ
(22)
with the universal two-factor scaling function Oðx; yÞ ¼ y4 ðy2 1Þ2 þ x2 ,
(22a)
and the critical dynamic exponent z0 ¼ b þ 2 ¼ 6.
(23)
It is easy to see that the characteristic frequency has the limiting behaviors o0c Gqz0 ;
qx0 b1,
o0c Gqz0 4 x4 0 ;
(24a)
qx0 51.
(24b)
Go back to expression (20) of the dynamic structure factor S0 ðq; oÞ we want to examine in time. To this end, we define Z þ1 do iot e S0 ðq; oÞ, S0 ðq; tÞ ¼ (25) 1 2p which is the Fourier transform of the dynamic structure factor in real space. A straightforward integration gives S0 ðq; tÞ ¼
Gqb o0c t e . o0c
(26)
The above formula calls the following comments. Firstly, the factor Gqb =o0c is nothing else but the initial static structure factor S 0 ðqÞ, relation (8). Secondly, at the spinodal temperature and at scales smaller than the mesh size x (where critical fluctuations occur), we find that the short-time behavior of the dynamic structure factor obeys the following scaling law S0 ðq; tÞ ¼ qZ0 2 f 0 ðqR0 ðtÞÞ;
ðZ0 ¼ 0; q1 5x0 ; t0 ! t0s Þ
(27)
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with the time-characteristic length R0 ðtÞ ¼ G1=z0 t1=z0 ;
z0 ¼ 6.
(28)
The scaling function f 0 ðxÞ reads z0
f 0 ðxÞ ¼ ex .
(29)
Then, at criticality and at small scales compared to the mesh size, the dynamic structure factor depends only on two kinds of lengths, which are the wavelength q1 and the time-characteristic length R0 ðtÞ. The latter can be understood as the size of domains alternatively rich in A and B-polymers at time t. Equating the two scales x0 and R0 ðtÞ gives some time-scale, t0 , necessary to the formation of microdomains of size x0 . We have 0 t0 ¼ G1 xz 0 ;
z0 ¼ 6.
(30)
t0
Then, can be regarded as a saturation time. In the following section, we shall analyze the time-dependence of the structure factor, when the critical fluctuations are strong enough. It is the case of low-molecular-weight polymer chains. This will be done using a scaling argument. 4. Dynamical scaling 4.1. Static structure factor To investigate the effects of the strong fluctuations of composition on the dynamic structure factor, we will need the critical behavior of the static structure factor, of which the behavior was extensively studied using a field-theoretical renormalization-group (RG) [8,21]. It will be instructive to recall the essential stages of the static study of MPS, when fluctuations of composition cannot be underestimated. To this end, one has introduced a field theory of free energy functional [8] ! ! Z Z 4 4 Y X 1 u0 d 2 2 H½jq ¼ ðt0 þ q þ C 0 q Þjq jq þ j qj , ð2pÞ dd (31) 2 q 4! q1 ...q4 i¼1 qi j¼1 where the short notation Z Z dd q ð2pÞd q
(31a)
is used. In the above expression, jq accounts for the Fourier transform of the rescaled local composition fluctuation jðrÞ, which is a scalar field defined on the d-dimensional Euclidean space Rd . There, q denotes the d-dimensional wave-vector, and the quantities t0 (‘‘squared mass’’), C 0 and u0 are the bare parameters. The first two parameters are those defined above, and u0 the coupling constant. The field j and the bare parameters of the model have the dimensions: ½j ¼ L1d=2 , ½t0 ¼ L2 , ½C 0 ¼ L4 , and ½u0 ¼ Ld4 , where L is some length. Note that the coupling constant u0 becomes marginal at the critical dimension d c ¼ 4 of the system. The free propagator of the model is Z dd q iq:r G0 ðr; t0 ; C 0 ; u0 Þ ¼ e S0 ðq; t0 ; C 0 ; u0 Þ, (32) ð2pÞd where S 0 ðq; t0 ; C 0 ; u0 Þ is the mean-field structure factor, relation (8). The basic quantities are the connected correlation functions. In this work, we consider only the connected two-point correlation function Gðr; t0 ; C 0 ; u0 Þ ¼ hjð0ÞjðrÞic ¼ hjð0ÞjðrÞi hjð0ÞihjðrÞi,
(33)
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whose Fourier transform is nothing else but the structure factor Sðq; t0 ; C 0 ; u0 Þ ¼ hjq jq ic , that is Z Sðq; t0 ; C 0 ; u0 Þ ¼ dd reiqr Gðr; t0 ; C 0 ; u0 Þ.
159
(34)
Here, r ¼ jrj denotes the module of the position-vector r. The theory has been developed through the perturbation series with respect to the coupling constant u0 , and exhibits short-range divergences that first necessitate a regulation procedure. In practice, one chooses the dimensional one [16,17], with the regulator ¼ 4 d. It has been shown [21] that the field-theory of action (31) is renormalizable at any order of the perturbation series up to the critical dimension d c ¼ 4 , and that, the necessary counterterms, to remove short-range divergences, have the same structure as those of the usual theory ðC 0 ¼ 0Þ. Therefore, no C 0 dependent counterterm is needed. For convenience, we choose the dimensional renormalization with the minimal substraction procedure, introduced by ’t Hooft and Veltman [22–24]. The renormalized action is given by Ref. [8] ! ! Z Z 4 4 Y X 1 um d R 2 2 4 2 R R R H½jq ¼ ðtm þ q þ Cm q Þjq jq þ j qj , ð2pÞ dd (35) 2 q 4! q1 q4 i¼1 qi j¼1 where jR is the renormalized field. The quantities t, C and u are the renormalized parameters corresponding to the bare ones t0 , C 0 and u0 , respectively. There, m is an arbitrary mass (’t Hooft mass), introduced in such a way that t, C and u be dimensionless irrespective of d. For instance, the scale mass m may be m ¼ a1 (a being the monomer size). The relations between bare and renormalized quantities write [21] R j ¼ Z 1=2 j j ;
u0 ¼ m Z u u;
t0 ¼ m2 Zt t, C 0 ¼ m4 Z1 j C.
(36a) (36b)
Here, Z j , Z t , and Z u , functions only of u and ¼ 4 d, are the j-field, mass, and coupling constant renormalization factors, respectively. These are Laurent series in and powers series in u, that is P Z ¼ 1 þ i;j aij ui j . Now, the two-point correlation function of interest renormalizes according to Ref. [21] Sðq; t0 ; C 0 ; u0 ; Þ ¼ Z j S R ðq; t; C; u; m; Þ.
(37)
The renormalized two-point correlation function SR , as a function of , is free from singularities at ¼ 0, at fixed renormalized parameters t, C and u. The RG-equation for the renormalized structure factor S R can be obtained utilizing the independence of the bare one S on the scale mass m. We simply recall the expected equation [8] q mqm þ W ðuÞ ½2 þ Zt ðuÞtqt ½4 Zj ðuÞCqC þ Zj ðuÞ SR ðq; t; C; u; m; Þ ¼ 0, (38) qu where W ðuÞ ¼ mqm j0 u is the Wilson function, and Zi ðuÞ ¼ mqm j0 ln Z i , with i ¼ t; j, are the exponent functions. The notation qm j0 means the partial m-derivative at fixed bare parameters t0 , C 0 and u0 . We recall that the function W ðuÞ vanishes at fixed point u , that is W ðu Þ ¼ 0. The limit u ! u of the exponent functions gives the standard critical exponents [16,17,25] n¼
1 ; 2 þ Zt
Z ¼ Zj ,
(39)
which characterize the behaviors of the thermal correlation length and the two-point correlation function (at small distances), respectively. A dimensional analysis implies [8] fqqq þ mqm d S gS R ðq; t; C; u; m; Þ ¼ 0
(40)
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with d S ¼ 2 the m-dimension of SR . Combining Eqs. (38) and (40) yields q qqq W ðuÞ þ ½2 þ Zt ðuÞtqt þ ½4 Zj ðuÞCqC Zj ðuÞ þ 2 S R ðq; t; C; u; m; Þ ¼ 0. qu
(41)
This RG-equation has been solved using the so-called characteristics method [16,17,25]. We simply sketch the essential results. First, it has been found [8] that the spinodal curve along which the structure factor diverges at q ¼ q , is described by the equation ts C 1=s
(42)
with the critical cross-over exponent s ¼ nð4 ZÞ. Its three-dimensional value is [8] s ’ 2:501 0:005.
(43a) 1 4
At two dimensions, knowing the exact values n ¼ 1 and Z ¼ [16,17], one has [8] s ¼ 15 4 ¼ 3:75.
(43b)
The spinodal equation (42) must be compared to the mean-field one, relation (7). The latter corresponds to the exponent s0 ¼ 2, since Z0 ¼ 0. The inequality s1 os1 0 traduces the fact that the mean-field theory used in Section 2 underestimates the strong fluctuations of composition near the spinodal point. Second, it has been demonstrated [8] from the RG-equation (41) that the maximum point q of the structure factor scales as q C n=s .
(44) 1 2
1
414,
we have q 4q0 For mean-field values n0 ¼ and Z0 ¼ 0, result (6) is then recovered. Since n=s ¼ ð4 ZÞ (mean-field value). This indicates that the size of microdomains x is smaller than the mean-field one x0 . Therefore, the mesh size is overestimated when it is calculated using mean-field theory. Finally, at small scales compared to the mesh size x ðq1 ox Þ, the RG-equation (41) reveals that the structure factor behaves according to Ref. [8] SðqÞqZ2 ;
ðq4q Þ.
(45)
The mean-field behavior (8c) is then recovered for Z0 ¼ 0. 4.2. Dynamic structure factor Now, we have all ingredients to study the scaling form for the critical dynamic structure factor. The latter obeys the following scaling law SðqÞ o Sðq; oÞ ¼ F , (46) oc ðqÞ oc ðqÞ where oc ðqÞ is the characteristic frequency. The above scaling form is readily checked for its mean-field homologous, by combining relations (20) and (21). The appearance of the ratio SðqÞ=oc ðqÞ, in the above formula, is natural, since integration of Sðq; oÞ over frequencies o must give the static structure factor SðqÞ. The scaling function F ðxÞ then satisfies Z þ1 dxF ðxÞ ¼ 1. (46a) 0
We suppose that the characteristic frequency oc ðqÞ writes as oc ðqÞ ¼ q2 LðqÞS1 ðqÞ
(47)
with a nontrivial static structure factor SðqÞ. As assumption, we admit that the mobility of strands is not affected by the fluctuations of composition. Thus, the Onsager transport coefficient LðqÞ also scales as LðqÞGq2 ðq1 ox Þ. This can be justified by the fact that LðqÞ is given by LðqÞ1=nðqÞ, where nðqÞ is the portion of one strand at scale q1 . In addition, we note that although the connected chains are of
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low-molecular-weight, their initial radius is Gaussian. Therefore, nðqÞq2 , and LðqÞ then obeys the above mentioned behavior, independently on the treatment nature of the problem (using mean-field theory or scaling). With the help of the static structure factor behavior (45), we find that the characteristic frequency scales as oc ðqÞGqz ;
q1 5x
(48)
with the dynamic critical exponent z ¼ 6 Z.
(49)
Using the best three-dimensional value of the standard exponent Z [16,17], we find that z ’ 5:969 0:001.
(49a)
In dimension 2, Z ¼ 14 [16,17], and we have the exact result z ¼ 23 4 ¼ 5:75.
(49b)
Formula (48) for the characteristic frequency calls the following commentaries: Firstly, since zo6, the strong fluctuations of composition give rise to high relaxation rates, in comparison with those obtained using mean-field theory. We can say that latter underestimates the relaxation rates. Secondly, we emphasize that, when it is correctly normalized, the scaling law for the characteristic frequency (48) is universal, independently on the chemical nature of the two polymers and values of the crosslinking density. Thirdly, for Z0 ¼ 0, the mean-field behavior is naturally recovered, relation (24a). In Fig. 1, we superpose the schematic variations of the characteristic frequency from mean-field theory and dynamical scaling, upon the wavenumber q. Now, we are interested in the dynamic structure factor in time, Sðq; tÞ, which is the Fourier transform of Sðq; oÞ, where frequency o and t are the conjugate variables. We find, at criticality and at small scales compared to the mesh size x , that Sðq; tÞ ¼ qZ2 f ðqRðtÞÞ,
(50)
100
80
ωc
60
40
20
0 1.0
1.5
2.0
2.5
3.0
q Fig. 1. Schematic variations of the characteristic frequency oc ðqÞ from mean-field (dashed line) and scaling (solid line), upon the wavenumber q.
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4
R (t)
3
2
1
0 0
2
4
6
8
10
Γt Fig. 2. Schematic variations of the time-characteristic scale RðtÞ from mean-field (dashed line) and scaling (solid line), upon the rescaled time Gt.
where f ðxÞ is a universal scaling function, and RðtÞG1=z t1=z ;
ðz ¼ 6 ZÞ
(51)
is the new time-characteristic scale. Therefore, the final time t behaves as t Gxz ,
(52)
which is manifestly smaller than the mean-field one t0 , because of the strong fluctuations of composition. Indeed, we have x ox0 and zoz0 ¼ 6. The evident inequality RðtÞ4R0 ðtÞ traduces the fact that the instabilities domains rapidly grow in time, in the presence of strong fluctuations of composition. Finally, in Fig. 2, we superpose the schematic variations of the time-characteristic scale RðtÞ from mean-field theory and dynamical scaling, upon the rescaled time Gt. 5. Conclusions We recall that the purpose of the present paper was the computation of the dynamic structure factor of a crosslinked polymer blend, when the latter is cooled down from a high initial temperature towards a final one very close to the spinodal point. To study physics, we used two theoretical approaches, namely mean-field theory and scaling argument. The former is valid for those mixtures where fluctuations of composition are not strong, as high-molecular-weight polymer chains. To see this, we recall that, according to de Gennes [26], the width of the critical region around the consolute point, where a nonclassical phase behavior is expected, scales as M 1 , with M the common molecular-weight of the two polymers of the mixture. Therefore, the de Gennes criterion suggests that the critical region becomes very narrow for high-molecular-weight polymer mixtures, so-that they can be studied within the framework of the mean-field theory (random field approximation [19]). It is not the case, however, for those low-molecular-weight polymer mixtures, as recently shown by small-angle neutron-scattering and dynamic light-scattering [27]. More precisely, measurements revealed the existence of a cross-over in the static susceptibility from mean-field to nonclassical behavior at T ’ T c þ 30 C, where T c is the spinodal temperature of the mixture. The conclusion is that, in low-molecular-weight polymer mixtures case, the width
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of the critical region is larger than usual, an then, one must observe a nonclassical critical phase behavior. Thus, the mean-field theory is no longer reliable for the description of these systems. For high-molecular-weight crosslinked polymer chains, we used an extended Van Hove theory, and determined an exact expression for the expected dynamic structure factor. At criticality and at small scales compared to the microdomains size, we found that the dynamic structure factor obeys a universal scaling law, which depends only on the wavenumber q and the time-characteristic length R0 ðtÞt1=z0 , with the mean-field dynamic exponent z0 ¼ 6. The latter was interpreted as the size of instabilities domains at 0 time t. A comparison between R0 ðtÞ and the mesh size x0 gives a final time t0 xz 0 ðz0 ¼ 6Þ, necessary to form microdomains alternatively rich in A and B-polymers. For low-molecular-weight crosslinked polymer chains, however, the fluctuations of composition are strong enough, and use was made of a scaling argument to study the short-time behavior of the dynamic structure factor. In particular, we found that the dynamic exponent z ’ 5:969 0:001, at three dimensions, and z ¼ 23 4, at dimension 2. We emphasize that, in any case, dynamics are of short-time type, because the instabilities growing is stopped at a final time directly proportional to a power factor of the mesh size. Finally, we point out that the above scaling laws for the dynamic structure factor can be recovered using a supersymmetric approach combined with field-theoretical RG. We emphasize that this method has been successfully applied to those critical systems (liquid mixtures, one-component fluid. . .) obeying the so-called dynamical model A [16,28–30]. Also, for model C introduced in this paper, this extension is possible (but not trivial). Such a question and other physical aspects are under consideration. Acknowledgments We are grateful to Professor M. Daoud for illuminating discussions and fruitful correspondence. One of us (M.B.) would like to thank the Laboratoire Le´on Brillouin (Saclay) and the Service de Physique The´orique (Saclay) for their kinds of hospitality during his regular visits.
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P.-G. de Gennes, J. Phys. Lett. 40 (1979) 69. A. Bettachy, A. Derouiche, M. Benhamou, M. Daoud, J. Phys. II (Paris) 1 (1991) 153. A. Derouiche, A. Bettachy, M. Benhamou, M. Daoud, Macromolecules 25 (1992) 7188. T.A. Vilgis, M. Benmouna, M. Daoud, M. Benhamou, A. Bettachy, A. Derouiche, Polym. Network Blends 3 (1993) 59. M. Benmouna, T.A. Vilgis, M. Daoud, M. Benhamou, Macromolecules 27 (1994) 1172. M. Benmouna, T.A. Vilgis, M. Benhamou, A. Babaoui, M. Daoud, Macromol.: Theory Simul. 3 (1994) 557. A. Bettachy, A. Derouiche, M. Benhamou, M. Benmouna, T.A. Vilgis, M. Daoud, Macromol.: Theory Simul. 4 (1995) 67. M. Benhamou, J. Chem. Phys. 102 (1995) 5854. D.J. Read, M.G. Brereton, T.C.B. McLeish, J. Phys. II (Paris) 5 (1995) 1679. A. Bettachy, Thesis, Hassan II-Mohammedia University, 1995. A. Derouiche, Thesis, Hassan II-Mohammedia University, 1995. M. Riva, V.G. Benza, J. Phys. II (Paris) 7 (1997) 285. M. Benhamou, A. Derouiche, A. Bettachy, J. Chem. Phys. 106 (1997) 2513. A. Derouiche, M. Benhamou, A. Bettachy, Eur. Phys. J. E 13 (2005) 353. R.M. Briber, B.J. Bauer, Macromolecules 21 (1988) 3296. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. C. Itzykson, J.-M. Drouffe, Statistical Field Theory, vols. 1 and 2, Cambridge University Press, Cambridge, 1989. Van Hove, Phys. Rev. 93 (1954) 249; Van Hove, Phys. Rev. 95 (1954) 1374. P.-G. deGennes, Scaling Concept in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953. M. Benhamou, Int. J. Mod. Phys. A 8 (1993) 2581. G. ’t Hooft, M. Veltman, Nucl. Phys. B 44 (1972) 189. G. ’t Hooft, Nucl. Phys. B 61 (1973) 455. J.C. Collins, Nucl. Phys. B 80 (1974) 341. D. Amit, Field Theory the Renormalization Group and Critical Phenomena, McGraw-Hill, New York, 1978.
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[26] P.-G. de Gennes, J. Phys. Lett. Paris 38 (1977) L-441; J.-F. Joanny, J. Phys. A 11 (1978) L-177; K. Binder, J. Chem. Phys. 79 (1983) 6387. [27] F.S. Battes, et al., Phys. Rev. Lett. 65 (1990) 1893. [28] J.C. Niel, Phys. Rev. B 37 (1988) 2156. [29] J.C. Niel, Europhys. Lett. 9 (1989) 415. [30] J.C. Niel, Thesis, Paris VI University, 1990.
Physica A 373 (2007) 153–164 www.elsevier.com/locate/physa
Dynamic structure factor of crosslinked polymer blends M. Benhamou, M. Chahid Laboratoire de Physique des Polyme`res et Phe´nome`nes Critiques, Faculte´ des Sciences Ben M’sik, B.P. 7955, Casablanca, Morocco Received 2 April 2006 Available online 5 June 2006
Abstract In this paper, we are interested in the critical behavior of the dynamic structure factor of a crosslinked polymer blend made of two chemically incompatible polymer A and B, when it is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. Since the critical fluctuations occur over distances smaller than the mesh size (microdomains size), x , the dynamic structure factor should be governed by a short-time behavior we want to determine. We demonstrate that the final time, t , necessary to the appearance of microdomains alternatively rich in A and B-polymers, scales as t xz , with z a dynamic critical exponent. The investigation of the dynamic structure factor is first achieved using a mean-field approach, based on an extended Van Hove theory, and second by a scaling argument. The Van Hove theory is valid as long as the fluctuations of composition can be underestimated. Within the framework of this model, we determine an exact form for the dynamic structure factor, and in particular, we find that the corresponding dynamic exponent is z0 ¼ 6. Second, the study is extended to the case of crosslinked polymer blends of low-molecularweight, where fluctuations of composition are strong enough near the spinodal temperature. Using a scaling argument, we prove that the scaling law for the dynamic structure factor is Sðq; tÞ ¼ qZ2 f ðqRðtÞÞ, with Z the standard Ising critical exponent, f ðxÞ a universal scaling function, and RðtÞt1=z some time-characteristic length. The latter can be interpreted as the size of instabilities domains at time t , and it becomes of the order of x at time t . The growing process of instabilities is then stopped at the final time t . We show that the dynamic exponent z is not trivial and has as three-dimensional value z ’ 5:969 0:001. In dimension 2, we find an exact value for this exponent that is z ¼ 23 4. r 2006 Elsevier B.V. All rights reserved. Keywords: Crosslinked polymer blends; Microphase separation; Dynamic structure factor
1. Introduction The physical system we consider is a mixture made of two polymers A and B of different chemical nature, crosslinked in their one-phase region by controlled irradiation. For the sake of simplicity, we assume that chains A are crosslinked to chains B, and there are no crosslinks between chains of the same kind. Thus, complications related to an eventual existence of crosslinks between chemically identical chains are not considered here. When the temperature of the system is lowered, the mixture has tendency to phase separate. Because of the presence of crosslinks, the elasticity of the network resists to such a separation. As result, the Corresponding author. Tel.: +212 22 704671; fax: +212 22 704675.
E-mail address: [email protected] (M. Benhamou). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.05.015
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crosslinked mixture undergoes a microphase separation (MPS), which is characterized by the appearance of microdomains alternatively rich in polymers A and B. The pioneered theory for studying the MPS was due to de Gennes [1], followed by several extended works [2–14]. The central idea of the author was an analogy between the crosslinked chains and a dielectric medium. The theoretical predictions was tested by Briber and Bauer [15] by small-angle neutron-scattering experiment on the PS-PVME mixture. The de Gennes’ theory agrees with experiment for all wave-vectors range except in the zero-scattering-angle limit. More precisely, the theoretical structure factor vanishes at this limit, while the experimental one does not. The discrepancy between theory and experiment was solved in a series of published papers [4,6,7,9]. Most works have been addressed to the static aspect of MPS. This means that only few studies were dedicated to the investigation of kinetics of such a transition [3,5,12,14], when the temperature is changed from an initial value T i towards a final one T f very close to the spinodal point. As noted in a previous work [3], the reptation motion of a cross-linked chain is frozen by the presence of permanent crosslinks. Hence, the only motion allowed to the cross-linked chains is of Rouse type. Kinetics was studied through the relaxation rate, as a function of the wave-vector. In this paper, we reexamine kinetics problem of MPS, in term of dynamic structure factor, when the system is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. The fact that the critical fluctuations occur over distances smaller than the mesh size (microdomains size), x , we expect that the dynamic structure factor is governed by a short-time behavior we want to determine. As we shall show below, the final time, t , necessary to the appearance of microdomains alternatively rich in A and B-polymers, scales as t xz , with z a dynamic critical exponent. This time-scale has a different meaning, when it is compared to the standard relaxation time concerning those critical systems exhibiting a phase transition occurring at vanishing wave-vector. The main difference comes from the fact that the latter diverges at the critical point [16,17], but t does not. Indeed, the relaxation time obeys a behavior similar to the above one (with another dynamic exponent), where x is replaced by the thermal correlation length xt . We shall demonstrate below, at the spinodal temperature and at scales smaller than the mesh size x , the leading critical behavior of the dynamic structure factor, Sðq; tÞ, depends only on two kinds of lengths, namely the wavelength q1 and RðtÞt1=z . The latter can be interpreted as the size of instabilities domains at time t, and it becomes of the order of x at time t . The growing process of instabilities is then stopped at the final time t . The investigation of dynamic structure factor will be first apprehended using a mean-field approach, based on an extended Van Hove theory [18]. Within the framework of this model, we compute exactly the dynamic structure factor, and in particular, we find that the corresponding dynamic exponent is z0 ¼ 6. Second, we shall extend the study to the case of an eventual presence of strong fluctuations of composition near the spinodal temperature. This will be done using a scaling argument. In particular, we find that the dynamic exponent z has as three-dimensional value z ’ 5:969 0:001. In dimension 2, we show that this exponent is exact and has the value z ¼ 23 4. The remainder of presentation proceeds as follows. In Section 2, we recall the essential of the static study of MPS within the framework of de Gennes theory. The exact computation of the dynamic structure factor, using mean-field theory, is the aim of Section 3. We examine, in Section 4, the dynamical scaling. Some concluding remarks are drawn in the last section.
2. De Gennes theory Consider a mixture of two chemically incompatible polymers A and B, cross-linked in the one-phase region. When the system is cooled down, below some critical temperature, a MPS takes place. Such a transition results from a competition between the usual phase separation and the elastic properties of the gel. Following de Gennes [1], the free energy functional of the system, in a Landau approximation, writes Z F0 ½j 1 ¼ (1) dd rft0 j2 þ ðrjÞ2 þ C 0 P2 g, kB T 2
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where T is the absolute temperature, kB the Boltzmann’s constant, and r 2 Rd the d-dimensional positionvector of the representative point of the system. The notation t0 ¼ a2 12ðwc wÞ
(2)
means the temperature parameter, where a is the monomer size, w the standard Flory interaction parameter, which is inversely proportional to the absolute temperature T, and wc ¼ 2=N its critical value if the system is uncrosslinked (N being the common polymerization degree of chains). The field or order parameter jðrÞ is directly proportional to the local fluctuation of composition rðrÞ ¼ FðrÞ Fc , that is jðrÞ ¼ a1d=2 rðrÞ. Here, F is the composition of one component of the mixture and Fc ¼ 12 its critical value. We recall that the first two terms in relation (1) constitute the expansion of the usual Flory–Huggins free energy [19,20] around the critical composition, and the third one describes the gel elasticity. The elastic contribution in this relation was first introduced by de Gennes by analogy with the polarization of a dielectric medium. The internal rigidity constant C 0 was related to the average number n of monomers per strand by Ref. [1] C 0 a4 n2 .
(3)
By strand, we mean the section of a chain between two consecutive crosslinks. The quantity 1=n was interpreted as the crosslinking density [1]. In equality (1), the vector P denotes the average displacement between the centres of masses of A and B strands; it is the analog of the polarization in the dielectric problem. The monomers A and B play the role of charges ðþÞ and ðÞ of the dielectric medium, respectively. The charge fluctuations correspond to the fluctuations of composition. The quantities P and r are not independent each other, but related by the Maxwell’s relation: div P ¼ rðrÞ. In reciprocal space, the latter writes: iq Pq ¼ rq , where rq ¼ a1þd=2 jq is the Fourier transform of the local fluctuation of composition rðrÞ ¼ a1þd=2 jðrÞ and q the wave-vector. In reciprocal space, the free energy (1) becomes Z F0 ½jq 1 2 dd qS1 ¼ (4) 0 ðqÞjjq j 2 kB T with S0 ðqÞ ¼
t0 þ
q2
1 þ C 0 q2
(5)
the structure factor. This differs from the usual one ðC 0 ¼ 0Þ by the extra term C 0 q2 taking into account the permanent presence of crosslinks. It exhibits a maximum at the finite value [1] 1=4
q0 ¼ C 0 , x0
(6) q1 0
pffiffiffi 1=4 C 0 a n,
x0
whose inverse, ¼ ¼ measures the size of microdomains A or B. The length is of the order of the size of strands (mesh size), that is x0 ¼ an1=2 . The structure factor S0 ðqÞ at q ¼ q0 diverges at the spinodal for Ref. [1] pffiffiffiffiffiffi t0s ¼ 2 C 0 . (7) The divergence of the structure factor at nonzero value of q is interpreted as a MPS [1]. The above spinodal equation tells us that the crosslinked mixture is more miscible than the uncrosslinked one. Finally, the structure factor can be rewritten on the following useful form S0 ðqÞ ¼
1 , 2 2 q2 ðq2 q2 0 Þ þ x0
(8)
with x0 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi t0 t0s
(9)
the thermal correlation length. Formula (8) suggests that, in the critical domain around t0 ¼ t0s and q ¼ q0 , the static structure factor depends on three kinds of parameters, which are the wavelength q1 , the microdomains
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size x0 and the thermal correlation length x0 . On the other hand, relation (8) can be written on the scaling form S0 ðqÞ ¼ qZ0 2 Xðqx0 ; qx0 Þ;
Z0 ¼ 0,
(8a)
with the universal two-factor scaling function Xðx; yÞ ¼
1 . y4 ðy2 1Þ2 þ x2
(8b)
In particular, at criticality and at scales smaller than the mesh size x0 , the static structure factor behaves as S0 ðqÞqZ0 2 ðZ0 ¼ 0Þ;
ðq1 5x0 ; t0 ! t0s Þ.
(8c)
The next step consists in the computation of the dynamic structure factor of a critical crosslinked polymer blend, using an extended Van Hove theory [18]. The latter is valid as long as we are concerned with highmolecular-weight polymer chains. 3. Extended Van Hove theory We start with a crosslinked polymer blend out of equilibrium. Denote by jðr; tÞ the local composition fluctuation of crosslinked A-chains. These, for example, diffuse with a local current J. The latter is related to jðr; tÞ by the standard continuity equation qjðr; tÞ þ r Jðr; tÞ ¼ 0. qt
(10)
Here, r is the position vector of the representative point of the medium and t the time. On the other hand, we recall the relationship between the current J and the chemical potential mðrÞ ¼ dF0 =djðrÞ (d=dj means the functional derivative), where F0 is the free energy (1), Z 1 dd r0 Lðr r0 Þrmðr0 Þ. (11) JðrÞ ¼ kB T Here, L means the Onsager transport coefficient to be discussed below. Now, combining relations (10) and (11) yields the Langevin equation Z qjðr; tÞ dF0 ¼ dd r0 Dr0 Lðr r0 Þ þ nðr; tÞ, qt djðr0 Þ
(12)
where D is the Laplacian operator. We have introduced a stochastic force (or noise) nðr; tÞ. It will be convenient to examine the problem in reciprocal space. Fourier transforming the above differential equation yields qjq ¼ q2 LðqÞðt0 þ q2 þ C 0 q2 Þjq þ nq ðtÞ. qt
(13)
This constitutes an extension of the well-known Van Hove equation (model C) [18]. The stochastic force is assumed to be Gaussian, that is hnq ðtÞnq0 ðt0 Þi ¼ 2q2 LðqÞdd ðq q0 Þdðt t0 Þ.
(14)
Equalities (13) and (14) show that physics depends on the knowledge of the Onsager transport coefficient LðqÞ. To specify its form, we first note that, due to the permanent reticulations of chains, only local motions are possible. Therefore, the Rouse motion of connected chains dominates, and then, the diffusion is essentially governed by free monomers. At small-scales, that is for q1 5x0 , LðqÞ behaves as LðqÞGq2 [3], where G is the kinetic coefficient. With these considerations, the basic equation becomes qjq ¼ Gqb ðt0 þ q2 þ C 0 q2 Þjq þ nq ðtÞ, qt
(15)
hnq ðtÞnq0 ðt0 Þi ¼ 2Gqb dd ðq q0 Þdðt t0 Þ
(16)
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with the exponent b ¼ 4. Notice that the model used here is slightly different from the usual ones describing nonconserved ðb ¼ 0Þ and conserved order parameters ðb ¼ 2Þ. We are interested in the dynamic structure factor defined by S0 ðr r0 ; t t0 Þ ¼ hjðr; tÞjðr0 ; t0 Þin .
(17)
We denote by S0 ðq; oÞ ¼ hjðq; oÞjðq; oÞin
(18)
its Fourier transform. The notation hin means the average over the stochastic force. Then, in the total reciprocal space, the solution to the motion equation (15) reads nq;o . (19) jq;o ¼ io þ Gqb ðt0 þ q2 þ C 0 q2 Þ The use of Eq. (16) leads to the dynamic structure factor S0 ðq; oÞ ¼
2Gqb o2 þ ðo0c Þ2
(20)
with b 2 2 o0c ¼ Gqb S1 0 ðqÞ ¼ Gq ðt0 þ q þ C 0 q Þ
(21)
the characteristic frequency, which defines the half-width at half-height. This frequency may also be rewritten on the scaling form o0c ¼ Gqz0 Oðqx0 ; qx0 Þ
(22)
with the universal two-factor scaling function Oðx; yÞ ¼ y4 ðy2 1Þ2 þ x2 ,
(22a)
and the critical dynamic exponent z0 ¼ b þ 2 ¼ 6.
(23)
It is easy to see that the characteristic frequency has the limiting behaviors o0c Gqz0 ;
qx0 b1,
o0c Gqz0 4 x4 0 ;
(24a)
qx0 51.
(24b)
Go back to expression (20) of the dynamic structure factor S0 ðq; oÞ we want to examine in time. To this end, we define Z þ1 do iot e S0 ðq; oÞ, S0 ðq; tÞ ¼ (25) 1 2p which is the Fourier transform of the dynamic structure factor in real space. A straightforward integration gives S0 ðq; tÞ ¼
Gqb o0c t e . o0c
(26)
The above formula calls the following comments. Firstly, the factor Gqb =o0c is nothing else but the initial static structure factor S 0 ðqÞ, relation (8). Secondly, at the spinodal temperature and at scales smaller than the mesh size x (where critical fluctuations occur), we find that the short-time behavior of the dynamic structure factor obeys the following scaling law S0 ðq; tÞ ¼ qZ0 2 f 0 ðqR0 ðtÞÞ;
ðZ0 ¼ 0; q1 5x0 ; t0 ! t0s Þ
(27)
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with the time-characteristic length R0 ðtÞ ¼ G1=z0 t1=z0 ;
z0 ¼ 6.
(28)
The scaling function f 0 ðxÞ reads z0
f 0 ðxÞ ¼ ex .
(29)
Then, at criticality and at small scales compared to the mesh size, the dynamic structure factor depends only on two kinds of lengths, which are the wavelength q1 and the time-characteristic length R0 ðtÞ. The latter can be understood as the size of domains alternatively rich in A and B-polymers at time t. Equating the two scales x0 and R0 ðtÞ gives some time-scale, t0 , necessary to the formation of microdomains of size x0 . We have 0 t0 ¼ G1 xz 0 ;
z0 ¼ 6.
(30)
t0
Then, can be regarded as a saturation time. In the following section, we shall analyze the time-dependence of the structure factor, when the critical fluctuations are strong enough. It is the case of low-molecular-weight polymer chains. This will be done using a scaling argument. 4. Dynamical scaling 4.1. Static structure factor To investigate the effects of the strong fluctuations of composition on the dynamic structure factor, we will need the critical behavior of the static structure factor, of which the behavior was extensively studied using a field-theoretical renormalization-group (RG) [8,21]. It will be instructive to recall the essential stages of the static study of MPS, when fluctuations of composition cannot be underestimated. To this end, one has introduced a field theory of free energy functional [8] ! ! Z Z 4 4 Y X 1 u0 d 2 2 H½jq ¼ ðt0 þ q þ C 0 q Þjq jq þ j qj , ð2pÞ dd (31) 2 q 4! q1 ...q4 i¼1 qi j¼1 where the short notation Z Z dd q ð2pÞd q
(31a)
is used. In the above expression, jq accounts for the Fourier transform of the rescaled local composition fluctuation jðrÞ, which is a scalar field defined on the d-dimensional Euclidean space Rd . There, q denotes the d-dimensional wave-vector, and the quantities t0 (‘‘squared mass’’), C 0 and u0 are the bare parameters. The first two parameters are those defined above, and u0 the coupling constant. The field j and the bare parameters of the model have the dimensions: ½j ¼ L1d=2 , ½t0 ¼ L2 , ½C 0 ¼ L4 , and ½u0 ¼ Ld4 , where L is some length. Note that the coupling constant u0 becomes marginal at the critical dimension d c ¼ 4 of the system. The free propagator of the model is Z dd q iq:r G0 ðr; t0 ; C 0 ; u0 Þ ¼ e S0 ðq; t0 ; C 0 ; u0 Þ, (32) ð2pÞd where S 0 ðq; t0 ; C 0 ; u0 Þ is the mean-field structure factor, relation (8). The basic quantities are the connected correlation functions. In this work, we consider only the connected two-point correlation function Gðr; t0 ; C 0 ; u0 Þ ¼ hjð0ÞjðrÞic ¼ hjð0ÞjðrÞi hjð0ÞihjðrÞi,
(33)
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whose Fourier transform is nothing else but the structure factor Sðq; t0 ; C 0 ; u0 Þ ¼ hjq jq ic , that is Z Sðq; t0 ; C 0 ; u0 Þ ¼ dd reiqr Gðr; t0 ; C 0 ; u0 Þ.
159
(34)
Here, r ¼ jrj denotes the module of the position-vector r. The theory has been developed through the perturbation series with respect to the coupling constant u0 , and exhibits short-range divergences that first necessitate a regulation procedure. In practice, one chooses the dimensional one [16,17], with the regulator ¼ 4 d. It has been shown [21] that the field-theory of action (31) is renormalizable at any order of the perturbation series up to the critical dimension d c ¼ 4 , and that, the necessary counterterms, to remove short-range divergences, have the same structure as those of the usual theory ðC 0 ¼ 0Þ. Therefore, no C 0 dependent counterterm is needed. For convenience, we choose the dimensional renormalization with the minimal substraction procedure, introduced by ’t Hooft and Veltman [22–24]. The renormalized action is given by Ref. [8] ! ! Z Z 4 4 Y X 1 um d R 2 2 4 2 R R R H½jq ¼ ðtm þ q þ Cm q Þjq jq þ j qj , ð2pÞ dd (35) 2 q 4! q1 q4 i¼1 qi j¼1 where jR is the renormalized field. The quantities t, C and u are the renormalized parameters corresponding to the bare ones t0 , C 0 and u0 , respectively. There, m is an arbitrary mass (’t Hooft mass), introduced in such a way that t, C and u be dimensionless irrespective of d. For instance, the scale mass m may be m ¼ a1 (a being the monomer size). The relations between bare and renormalized quantities write [21] R j ¼ Z 1=2 j j ;
u0 ¼ m Z u u;
t0 ¼ m2 Zt t, C 0 ¼ m4 Z1 j C.
(36a) (36b)
Here, Z j , Z t , and Z u , functions only of u and ¼ 4 d, are the j-field, mass, and coupling constant renormalization factors, respectively. These are Laurent series in and powers series in u, that is P Z ¼ 1 þ i;j aij ui j . Now, the two-point correlation function of interest renormalizes according to Ref. [21] Sðq; t0 ; C 0 ; u0 ; Þ ¼ Z j S R ðq; t; C; u; m; Þ.
(37)
The renormalized two-point correlation function SR , as a function of , is free from singularities at ¼ 0, at fixed renormalized parameters t, C and u. The RG-equation for the renormalized structure factor S R can be obtained utilizing the independence of the bare one S on the scale mass m. We simply recall the expected equation [8] q mqm þ W ðuÞ ½2 þ Zt ðuÞtqt ½4 Zj ðuÞCqC þ Zj ðuÞ SR ðq; t; C; u; m; Þ ¼ 0, (38) qu where W ðuÞ ¼ mqm j0 u is the Wilson function, and Zi ðuÞ ¼ mqm j0 ln Z i , with i ¼ t; j, are the exponent functions. The notation qm j0 means the partial m-derivative at fixed bare parameters t0 , C 0 and u0 . We recall that the function W ðuÞ vanishes at fixed point u , that is W ðu Þ ¼ 0. The limit u ! u of the exponent functions gives the standard critical exponents [16,17,25] n¼
1 ; 2 þ Zt
Z ¼ Zj ,
(39)
which characterize the behaviors of the thermal correlation length and the two-point correlation function (at small distances), respectively. A dimensional analysis implies [8] fqqq þ mqm d S gS R ðq; t; C; u; m; Þ ¼ 0
(40)
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with d S ¼ 2 the m-dimension of SR . Combining Eqs. (38) and (40) yields q qqq W ðuÞ þ ½2 þ Zt ðuÞtqt þ ½4 Zj ðuÞCqC Zj ðuÞ þ 2 S R ðq; t; C; u; m; Þ ¼ 0. qu
(41)
This RG-equation has been solved using the so-called characteristics method [16,17,25]. We simply sketch the essential results. First, it has been found [8] that the spinodal curve along which the structure factor diverges at q ¼ q , is described by the equation ts C 1=s
(42)
with the critical cross-over exponent s ¼ nð4 ZÞ. Its three-dimensional value is [8] s ’ 2:501 0:005.
(43a) 1 4
At two dimensions, knowing the exact values n ¼ 1 and Z ¼ [16,17], one has [8] s ¼ 15 4 ¼ 3:75.
(43b)
The spinodal equation (42) must be compared to the mean-field one, relation (7). The latter corresponds to the exponent s0 ¼ 2, since Z0 ¼ 0. The inequality s1 os1 0 traduces the fact that the mean-field theory used in Section 2 underestimates the strong fluctuations of composition near the spinodal point. Second, it has been demonstrated [8] from the RG-equation (41) that the maximum point q of the structure factor scales as q C n=s .
(44) 1 2
1
414,
we have q 4q0 For mean-field values n0 ¼ and Z0 ¼ 0, result (6) is then recovered. Since n=s ¼ ð4 ZÞ (mean-field value). This indicates that the size of microdomains x is smaller than the mean-field one x0 . Therefore, the mesh size is overestimated when it is calculated using mean-field theory. Finally, at small scales compared to the mesh size x ðq1 ox Þ, the RG-equation (41) reveals that the structure factor behaves according to Ref. [8] SðqÞqZ2 ;
ðq4q Þ.
(45)
The mean-field behavior (8c) is then recovered for Z0 ¼ 0. 4.2. Dynamic structure factor Now, we have all ingredients to study the scaling form for the critical dynamic structure factor. The latter obeys the following scaling law SðqÞ o Sðq; oÞ ¼ F , (46) oc ðqÞ oc ðqÞ where oc ðqÞ is the characteristic frequency. The above scaling form is readily checked for its mean-field homologous, by combining relations (20) and (21). The appearance of the ratio SðqÞ=oc ðqÞ, in the above formula, is natural, since integration of Sðq; oÞ over frequencies o must give the static structure factor SðqÞ. The scaling function F ðxÞ then satisfies Z þ1 dxF ðxÞ ¼ 1. (46a) 0
We suppose that the characteristic frequency oc ðqÞ writes as oc ðqÞ ¼ q2 LðqÞS1 ðqÞ
(47)
with a nontrivial static structure factor SðqÞ. As assumption, we admit that the mobility of strands is not affected by the fluctuations of composition. Thus, the Onsager transport coefficient LðqÞ also scales as LðqÞGq2 ðq1 ox Þ. This can be justified by the fact that LðqÞ is given by LðqÞ1=nðqÞ, where nðqÞ is the portion of one strand at scale q1 . In addition, we note that although the connected chains are of
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low-molecular-weight, their initial radius is Gaussian. Therefore, nðqÞq2 , and LðqÞ then obeys the above mentioned behavior, independently on the treatment nature of the problem (using mean-field theory or scaling). With the help of the static structure factor behavior (45), we find that the characteristic frequency scales as oc ðqÞGqz ;
q1 5x
(48)
with the dynamic critical exponent z ¼ 6 Z.
(49)
Using the best three-dimensional value of the standard exponent Z [16,17], we find that z ’ 5:969 0:001.
(49a)
In dimension 2, Z ¼ 14 [16,17], and we have the exact result z ¼ 23 4 ¼ 5:75.
(49b)
Formula (48) for the characteristic frequency calls the following commentaries: Firstly, since zo6, the strong fluctuations of composition give rise to high relaxation rates, in comparison with those obtained using mean-field theory. We can say that latter underestimates the relaxation rates. Secondly, we emphasize that, when it is correctly normalized, the scaling law for the characteristic frequency (48) is universal, independently on the chemical nature of the two polymers and values of the crosslinking density. Thirdly, for Z0 ¼ 0, the mean-field behavior is naturally recovered, relation (24a). In Fig. 1, we superpose the schematic variations of the characteristic frequency from mean-field theory and dynamical scaling, upon the wavenumber q. Now, we are interested in the dynamic structure factor in time, Sðq; tÞ, which is the Fourier transform of Sðq; oÞ, where frequency o and t are the conjugate variables. We find, at criticality and at small scales compared to the mesh size x , that Sðq; tÞ ¼ qZ2 f ðqRðtÞÞ,
(50)
100
80
ωc
60
40
20
0 1.0
1.5
2.0
2.5
3.0
q Fig. 1. Schematic variations of the characteristic frequency oc ðqÞ from mean-field (dashed line) and scaling (solid line), upon the wavenumber q.
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4
R (t)
3
2
1
0 0
2
4
6
8
10
Γt Fig. 2. Schematic variations of the time-characteristic scale RðtÞ from mean-field (dashed line) and scaling (solid line), upon the rescaled time Gt.
where f ðxÞ is a universal scaling function, and RðtÞG1=z t1=z ;
ðz ¼ 6 ZÞ
(51)
is the new time-characteristic scale. Therefore, the final time t behaves as t Gxz ,
(52)
which is manifestly smaller than the mean-field one t0 , because of the strong fluctuations of composition. Indeed, we have x ox0 and zoz0 ¼ 6. The evident inequality RðtÞ4R0 ðtÞ traduces the fact that the instabilities domains rapidly grow in time, in the presence of strong fluctuations of composition. Finally, in Fig. 2, we superpose the schematic variations of the time-characteristic scale RðtÞ from mean-field theory and dynamical scaling, upon the rescaled time Gt. 5. Conclusions We recall that the purpose of the present paper was the computation of the dynamic structure factor of a crosslinked polymer blend, when the latter is cooled down from a high initial temperature towards a final one very close to the spinodal point. To study physics, we used two theoretical approaches, namely mean-field theory and scaling argument. The former is valid for those mixtures where fluctuations of composition are not strong, as high-molecular-weight polymer chains. To see this, we recall that, according to de Gennes [26], the width of the critical region around the consolute point, where a nonclassical phase behavior is expected, scales as M 1 , with M the common molecular-weight of the two polymers of the mixture. Therefore, the de Gennes criterion suggests that the critical region becomes very narrow for high-molecular-weight polymer mixtures, so-that they can be studied within the framework of the mean-field theory (random field approximation [19]). It is not the case, however, for those low-molecular-weight polymer mixtures, as recently shown by small-angle neutron-scattering and dynamic light-scattering [27]. More precisely, measurements revealed the existence of a cross-over in the static susceptibility from mean-field to nonclassical behavior at T ’ T c þ 30 C, where T c is the spinodal temperature of the mixture. The conclusion is that, in low-molecular-weight polymer mixtures case, the width
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of the critical region is larger than usual, an then, one must observe a nonclassical critical phase behavior. Thus, the mean-field theory is no longer reliable for the description of these systems. For high-molecular-weight crosslinked polymer chains, we used an extended Van Hove theory, and determined an exact expression for the expected dynamic structure factor. At criticality and at small scales compared to the microdomains size, we found that the dynamic structure factor obeys a universal scaling law, which depends only on the wavenumber q and the time-characteristic length R0 ðtÞt1=z0 , with the mean-field dynamic exponent z0 ¼ 6. The latter was interpreted as the size of instabilities domains at 0 time t. A comparison between R0 ðtÞ and the mesh size x0 gives a final time t0 xz 0 ðz0 ¼ 6Þ, necessary to form microdomains alternatively rich in A and B-polymers. For low-molecular-weight crosslinked polymer chains, however, the fluctuations of composition are strong enough, and use was made of a scaling argument to study the short-time behavior of the dynamic structure factor. In particular, we found that the dynamic exponent z ’ 5:969 0:001, at three dimensions, and z ¼ 23 4, at dimension 2. We emphasize that, in any case, dynamics are of short-time type, because the instabilities growing is stopped at a final time directly proportional to a power factor of the mesh size. Finally, we point out that the above scaling laws for the dynamic structure factor can be recovered using a supersymmetric approach combined with field-theoretical RG. We emphasize that this method has been successfully applied to those critical systems (liquid mixtures, one-component fluid. . .) obeying the so-called dynamical model A [16,28–30]. Also, for model C introduced in this paper, this extension is possible (but not trivial). Such a question and other physical aspects are under consideration. Acknowledgments We are grateful to Professor M. Daoud for illuminating discussions and fruitful correspondence. One of us (M.B.) would like to thank the Laboratoire Le´on Brillouin (Saclay) and the Service de Physique The´orique (Saclay) for their kinds of hospitality during his regular visits.
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