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Interpolating between quenched and annealed disorder
PHYSICA ELSEVIER
Physica A 211 (1994) 365-380
Interpolating between quenched and annealed disorder Michael E Zimmer 1 Department of Physics, Materials Research Laboratory, and Beckman Institute, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, 1L 61801-3080, USA
Received 30 August 1993; revised 1 July 1994
Abstract A host-impurity system is considered, and the effect on the host due to a slow relaxation of the impurity is studied, when each can be described by a continuous field. Langevin dynamics are assumed for each field, along with a special choice of initial conditions that corresponds to a quenched equilibrium. Equal-time correlation functions interpolate between the limiting cases of a quenched and annealed impurity. An effective critical temperature may be defined, which is dependent on the relaxation rate of the impurity and the duration of the experiment. A simulation for the case of a conserved impurity and nonconserved host is used to demonstrate how a coexistence curve can become smeared when the relaxation rate of the impurity is small. The impurity enslaves the host, causing it to phase separate with a pattern that is more reminiscent of a conserved field. Also, it is suggested that the approach used could shed some light on the difficulties with one of the fixed points of Model C.
1. Introduction Impurities are typically classified as either quenched (i.e. relatively immobile) or annealed (i.e. relatively free to move about). While this is useful for describing most cases, it does not account for those impurities that can only partially anneal on the time scales o f the experiment. A possible example could be realized in SrTiO3, where slowly relaxing defects are considered to be crucial for an understanding o f the central peak in its displacive structural phase transition [ 1 ]. Also, a gel immersed in a binary fluid has been viewed as a quenched random impurity problem [2]. However, in such I Current address: IFF, Forschungszentrum Jiilich, D-52425 Jiilich, Germany. 0378-4371/94/$07.00 (~) 1994 Elsevier Science B.V. All fights reserved SSD10378-4371 (94)00178-2
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a system the polymers (impurity) are free to move around to some degree, and it has been wondered how to quantify the effect of the slow relaxation of the polymers on the phase separation of the fluid [3]. In addition, there have been studies [4] treating the slow annealing of impurities, e.g. the annealing of the Cr 5+ impurities substituted for As in KD2AsO4. However, the long-time annealing of the Cr 5+ was not considered and so any discussion of an interpolation becomes precluded. Besides these, other works treating annealed impurities tended to concentrate on the asymptotic behavior, rather than on any kind of a connection between the two equilibrium averages. These studies dealt with surfactants and polymer blends; their similarities will be pointed out in turn. In this paper we shall investigate the effect of a slow impurity, first by studying a simple model above the (equilibrium) annealed critical temperature (Tc(a)), and then by simulating a similar model below To(a). Above Tc(a), we give correlation functions that interpolate between the quenched and annealed cases. This is done for the simple but illustrative case of a Gaussian model; it is sufficient to reveal the qualitative behavior we seek (and we believe that adding higher order terms would only complicate the model without adding any new physics). We can define an effective critical temperature, Teff, that ranges between the (equilibrium) quenched critical temperature (T~(q)) and Tc(a), being closer to Tc(q) when the relaxation rate of the impurity,/'~, is smaller. Representative plots of Teyf vs. F~ and Teff vs. duration of experiment, ~', are given for comparison with numerical experiment. Below the annealed critical temperature it is more useful to proceed via simulation, as direct analytical approaches are less effective. Using a cell dynamical scheme [5] we simulate, as an example, a nonconserved host and a conserved impurity below the annealed critical temperature. A smearing of the coexistence curve about an effective critical temperature is demonstrated. In this case the host is enslaved by the impurity, and the phase separation is morphologically similar to that of a conserved ho.~t. In some models there arises the case where the relaxation rate of one of the fields vanishes; such is the case at one of the renormalization group fixed points of Model C [6,7]. In studying the stability of these fixed points, small but finite relaxation rates must be allowed for. However, this is seen to have the effect of making the field act annealed, when it was originally quenched at the fixed point. This is mentioned as being a possible reason for the ambiguities arising with this fixed point for this model. In Section 2 the statistics of the limiting cases of quenched and annealed impurities are reviewed. A dynamical interpolation is argued for and the necessary experimental scenario discussed. In Section 3 dynamics are introduced via Langevin equations in order to calculate an explicit interpolation on a model bilinear Hamiltonian. Among other things, graphical dependences are found for an effective critical temperature vs. the impurity relaxation rate, as well as the total time of relaxation. In Section 4 the particular case of a nonconserved host interacting with a conserved impurity is simulated below its annealed critical temperature. The evolution of ¢2 (~p = host field) integrated over the system is studied as a function of temperature and duration of quench. It is meant to provide a measure of the phase separation and is in qualitative agreement with the results of the bilinear model. In Section 5 the relevance of the simultaneous limits
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to ~ 0, ? ~ cx~ to renormalization group fixed points at to = 0 is discussed (to = relaxation rate, ? = duration of relaxation). Finally, concluding remarks are made in Section 6.
2. General considerations To make progress using the standard techniques, it is necessary to assume a Hamiltonian High, ~o], which determines the joint probability distribution of the host ((b) and the impurity (~o). As this Hamiltonian is supposed to determine the static behavior, it holds equally true in the case when the dynamics of the impurity are frozen. Thus, for H[~b,~o] polynomial in its arguments, we define HI,o] -- H[0,~o] (which represents self-~p interactions) and set H[~p,~o] = n[~bl~o] + H[~o];
(2.1)
H[~bl~o] is the Hamiltonian for interactions of ~0 given a fixed distribution of ~o. It must not be forgotten that this holds true at only a bare (unrenormalized) level, since a quenched and annealed Hamiltonian will in general be renormalized differently. The above three H ' s are all that is needed to calculate correlation functions in the two equilibrium limits. First, in the quenched limit, when the host has relaxed about a frozen impurity, the equilibrium correlations of ~b may be determined from the free energy ( F ) :
F=-fT~ozP[~oi]ln{fT~Oe-m*l~'l},
(2.2)
where P [ ~oI] is the probability distribution for ~'1, and D o and D~Ol represent integrations over all field configurations, as usual. Were the impurity instead able to explore the host over some time scale (as set by experiment), then we would rather calculate the free energy, using
F = - In {fZ~0~9,,
e-O~C"*t~}.
(2.3)
Given that the above two expressions for F correspond to different limits of the impurity mobility, one may ask whether it is possible to form an interpolation between them. While there are a number of artificial interpolations that may be invented, a physical one presents itself. The scenario is to take the initial condition of ~p relaxed about a quenched ~o, and then to use dynamics that evolve ~p and ~o towards a mutual (annealed) equilibrium. Away from criticality, where relaxations typically behave as e -°'r (w = relaxation rate, ?= duration of relaxation), it is clear that the quenched and annealed (static) averages are recovered (from the equal-time correlation function) in the limits w? ~ 0 and w~"---, c~, respectively. Finally, because to is normally fixed by other parameters, we may just consider the time ? as the interpolation parameter. On a mathematical level it is clear that the above provides for an interpolation. In applying this to real systems however, our requisite initial condition forces us to focus
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only on cases where the impurity relaxes much more slowly than the host. That is, in preparing the system with a temperature jump, we require that after some time the host is effectively annealed and the impurity still effectively frozen. To assure this comfortably would require at least (roughly) two orders of magnitude difference in the relaxation rates of ~p and ~p. A forced application of the formulas would necessitate "holding down" the impurity until the host relaxed, and then letting go at the initial time. Finally, we note that the difficulties in applying these equations in intermediate cases mirrors the difficulty in choosing how to define an average of an evolving system.
3. Disordered phase In this Section we discuss for a particular example the interpolation between quenched and annealed correlation functions above T¢ (a), the annealed critical temperature. If the dynamics of the host and impurity are known, it is always possible to make such an interpolation, at least in principle. In the disordered phase and away from the quenched and annealed critical temperatures, a Gaussian model suffices to reveal the main qualitative features of an interpolation. Higher order terms could be included, but at this stage they would only complicate the picture. This model could be relevant in the disordered phase for those defects which, upon fluctuating, locally break the symmetry of the host. The defects are assumed to be able to (at least eventually) explore their entire phase space (i.e. ergodicity is not broken). For the sake of example we consider the case
U[~,, ~,1 = f d a x
{~ [r0~ 2 + ( V ~ ) 2] + ½ [r0~,~ + ( V ~ , ) ~] + u O ~ l } .
(3.1)
There are energy penalties for self-interactions and gradients of each field; there is also an interaction term between the two fields. 7-0, ro and v0 are positive constants. By inspection, the correlation for ~b-fluctuations is (~l,q = f eiqx~b(x)dx) (~Jq~.l_q)
=
(,)(~ 1 - u2 )(tp) - 1
,
(3.2)
showing annealed critical temperature Tc ( a ) = Tw + v2/7-0; we define X g, = ( ro + q2) -1 and X~ = ( t o + q 2 ) -1- For vo > 0 ( < 0), a positive fluctuation in ~b will favor a negative (positive) fluctuation in ~t, which in turn will favor a positive fluctuation in ~,. So regardless of the sign of vo, the interaction enhances correlations in ~b, as well as ~Pt (this follows from the symmetry of the two fields in Eq. (3.1)). For the impurity (with q~tq = f eiqxq~l (X) ), (~Olqq~l_q) = ( X ; 1 -- U2X,) -1 , f r o m w h i c h w e m a y identify an effective 7-, 7-eff = 7-0 _ U2/FO. AS r 0 ~
(3.3) O, r e f f m a y
become negative, indicating a phase transition. This "annealed" phase transition happens
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at the same temperature for both fields, when ro~'0 = v~. Of course, this model is unstable in this case because there are no higher order stabilizing terms (such as ~4 or q~4). In the case when the impurity is quenched, the appropriate Hamiltonian for the host ~ (for a given impurity configuration q~t) would be the same as Eq. (3.1) except without the quadratic terms in q~t. In the quenched case, because the interaction is bilinear, there is a change in ( ~ ) only because of a change in (~); it is easily seen 2 2 that (g/q¢_q) = X¢, + OoX~X¢,. Because of its quenched nature, correlations of ~oi will not change due to interactions with ~. The self-q~t interactions in F_x1. (3.1) immediately give (~Olq~Ot--q) .~ X~o" For the purpose of describing the interpolation, any dynamics may be used, but here we choose the following Langevin equations out of simplicity and also because they are useful in many physical models. In this example the equations are for nonconserved fields (i.e. f d d x all(x, t) is allowed to change; the simplest way to model conserved fields is to change F --~ F q 2) 0e~, = - F ~
0,~Ol = - F ~ ,
6 H [ ~9, ~Ol] 8~0 + v,
(3.4)
8 H [ ¢ , ~ot ] 6~Ol + ('
(3.5)
with zero-mean Gaussian white noise (~,( x, t ) u( x', t') ) = 2 F c B ( x - x ' ) 8 ( t - t'),
(3.6)
and similarly for (; ( ) denotes a noise average. There are no correlations between ~, and (. The initial conditions ¢ ( x , 0 ) -= Ct0(x), ~oi(x,O) - q~lo(x) are set with the weight e -H[O°l~'°] , where Mt
ol
,o] =
fd"x(
[ro~b ' d2 + (Vrpo) 2] +
.
(3.7)
(the meaning of the primed constants is explained below). This is the interaction energy for the ~9 field for a given impurity field gOlO (at t = 0). As it is quenched at t = 0, the impurity must be averaged over with the probability distribution at that time, which is denoted by P [ (Pl0] : P[goi0] o( exp
-
dax [TOq~;O+ (Vq~,o) 2]
)
=
.
(3.8)
The form for the above two equations is determined from H[~O, q~t ], the Hamiltonian for interactions of impurity and host, which gives the energy of self-interactions between fields. It is appropriate to consider it as being composed of a piece with only selfinteractions of q~l, H[q~l], and the piece H[O I ~ot], which was already defined. That is, at t = 0 we set (as previously mentioned) H[O,q~t] = H[¢o I ~OlO] + H[~OlO].
(3.9)
That we can write such an expression at the bare (unrenormalized) level is obvious. However, when higher-order interactions are present, corresponding parameters on the
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right and left hand sides of this equation will be affected differently; quenched and annealed interactions renormalize parameters in different ways. It is for this reason that the parameters which appear in Eqs. (3.7) and (3.8) even though are primed, they originally represented the same physical interactions as in Eq. (3.1). This is done merely as a bookkeeping device. Also, it is possible to write this in terms of a generating functional, which serves as a basis from which one can discuss its renormalizability, in a field-theoretic fashion 2. A simple calculation yields the following cumulants, for T > T¢(a) 3
FO (~bq(t)~b_q(t))= ( a + - - ' l _ )
[ A+ - r°O A_ (1
Xo¢
(q~lq(t)q~l_q(t)) c -
e-2a_t)
-
A- - ~aO A+
(1 - e
e_a+t
F¢/ (A+~-A_)
[A+-~o¢, ~-+
(1 -
-2a+t)
]
2, (3.14)
e_2~+t)
A_ - o)o ~ A_
(1 - e - 2 a - t )
~oo)--__ ("-~-? a _ - a, o) [e -'~+t - e-A-t] 2, -XO0 (a+ --'~+
]
(3.15)
where the effective relaxation rates are a± = ½
+
i ½
+44r ,r ,
-
(3.16)
and oJ0 = F~(ro + q2) and oJ¢, = F¢,(ro + q2) [14]. This correlation function is for a nonconserved host and a nonconserved impurity. X0~ comes from Eq. (3.5) and is 2 The generating functional for our problem is
W[J, K]
= [
J
Y)~otP[cp/0] In W[J, K
I ~,0]
(3.10)
9
W[J, Kl,,o]=/79¢7~,790oe-nt*ol~'o~6[O-O~lSt~1-,,clexp(fdaxdt(JO+K,l)). (3.11) One can argue that there will not be any new divergences induced by this formalism. This is aside from the short time divergences that result from when one of the external points of certain correlation functions approaches the initial time surface. The parameters and fields that are restricted to the initial time surface will be renormalized as in the corresponding quenched problem. The renormalizations of parameters off the surface seems to be the same as in the bulk theory (i.e. the annealed case). For a discussion of this see [8]. 3 The usual correlation functions are given by: u2 ~ v2
(#) = ( # ) c
+
oXu~oa¢
(~-
~-2-)2
[A_e-a+ ' - A + e - ~ - ' ] 2
o~X~ r, [a_(x+ - ~oo)e-;t+t (~,~1) = (¢,~,)c + (,~+X--r--,2
(3.12)
A+(A_ - o ¢ , ) e - a - ' ]~2 .
(3.13)
M.F. Zimmer / Physica A 211 (1994) 365-380 20
20
15
15
-.~1o
10
371
V
5 0
5 I
i
-I .0
-0.5
I
0.0
0.5
.0
0 -I .0
-0.5
0.0
0.5
1.0
q
Fig. 1. (~.np)c is plotted with the values r0 = 0.15, ~'0 = 1, v2 = 0.1, re = 4., and F,t = 0.1 for: (a) nonconserved host, nonconserved impurity at times 0, 10, 100 ; (b) conserved host (i.e. F¢ ---} Fcq2), nonconserved impurity at times 0, 100, 1000, I0000. As time evolves the plots move upwards. equal to (r~ + q2) -1. However, since ro and r~ represent the same physical interaction,
Xo¢ = (ro + q2) -1. For T < Tc(a) this can only be used for short times, since this model does not have the necessary higher order stabilizing terms. It is easy to check that in the limits t ---} 0 and t ---* c~, the correlation functions arising from a quenched and annealed impurity are recovered, respectively. Finally, in the case o f F , I = 0, the impurity is quenched for all time, and it is straightforward to show that this formulation reduces down to the quenched impurity case studied by De Dominicis [9]. For T > Tc(a) the interpolating function (~kq(t)~_q(t)) c is plotted in Fig. 1 for the two cases o f a conserved (to~ = Fcq2(ro + q2)) and nonconserved (toe = F¢(ro + q2)) host with a nonconserved (¢o, = F~,(~'o + q2)) impurity. The graphs depict the evolution o f (~kq(t)~l_q(t)) c from t = 0, where the impurity distribution is quenched, to values large enough that the impurity distribution is effectively annealed. In Fig. l b the host is conserved, and thus (~q(t)~k_q(t)) c cannot change at q = 0. The global conservation leads to a selected momentum scale 4 which approaches zero as time increases. A t larger times it will most likely be difficult to observe since scattering experiments involve only a small subvolume (i.e. an open system), which are not constrained by conservation laws. Such a narrowing peak is reminiscent o f the structure factor in spinodal decomposition [ 10], where it reflects the dominant role played by the correlation length in bringing about dynamical scaling. In studies involving an annealing impurity (none o f which involved energy penalties for impurity gradients) this peak would saturate [ 11 ], reflecting the maximum size the correlation length could grow to on experimental time scales. Finally, the other two combinations, conserved impurity and conserved or nonconserved host, lead to interpolating functions that are qualitatively very similar to Fig. l b and so are not included 5 . 4 This length scale was found not to scale for times up to ,-, 10 9, with all constants ,~ O( 1). 5 The fact that there is a "pinning" at q = 0 when the host is nonconserved and the impurity is conserved can be shown to be due to the Gaussian nature of the theory (i.e., higher order terms eliminate this).
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M.F. Zimmer / Physica A 211 (1994) 365-380 0.1
,
,.<' - 0 . 2
.'" -"
..'/
- ....
•
P~,=O P~=.01
P~=.l
P~= 1
-0.5 -0.1
0.2
0.5
0.8
%
Fig. 2. The effective relaxation rate A- for a nonconserved host and impurity is plotted against ro = T - Tco for F¢ = 1, v2 = 0.6, ro = 1, q = 0 and F , = (i) 0, (ii) 0.01, (iii) 0.1 and (iv) 1. The intersection of -or with the A curves occurs at lffcff - Tco, by definition. The dotted line is - a . If f is defined as the duration of the experiment, then above Tc(a) quenched and annealed behavior will be observed when A_3 << c and A_3 >> c, respectively (assuming A+t >> c, where c is an O ( 1 ) constant). This defintion will necessarily include a qdependence 6 ; here, q = 0 is chosen. Below Tc(a) phase separation can only be observed if there has been sufficient (exponential) growth. Above Tc(q), the quenched critical temperature, this amounts to - a _ 3 > c, since only A_ can be negative here. So given 3, phase separation will only be observed if - A _ > c/? =_ or. In Fig. 2 we plot A_ for several values o f F~,, which intersect a representative - a line. By definition, the intersections take place at r0 = Te l f - Tco, which in turn leads to the dependence o f the effective Tc on the relaxation rate o f the impurity (see Fig. 3). As defined, Te l f ranges from Tc(q) to T~(a), up to a small shift that is due to a finite 3; thus we confine our attention to large L This effective Tc correctly changes from ~ T~(q) (for F,o,/F¢, ~ O) over to ~ Tc(a) (for F ~ J F ¢ ~ O ( 1 ) ) , and tends to be closer to To(a) for larger L If 3 --~ cx~ (i.e. at equilibrium) then Teyf would equal Tc(a) everywhere, except where F~, = 0, as must be the case. The sensitivity o f T[ y / o n the time 3 is demonstrated in Fig. 4. It gives a quantitative measure of how the absolute value o f the effective Tc's change, as well their values relative to other T [ f f ' s at a given relaxation rate. If in a series of experiments we critically quenched to several temperatures between Tc(a) and Tc(q) and waited a time L we would have several values o f A_3 and would thus see different stages of development of the phase separation. In this sense the coexistence curve is smeared over a range of temperature. This type of effect has been observed near the phase transition point of binary fluids in gels [3]. However, one can also argue [ 12] that the impurity field induces long lived metastable states in the host field. In this case it may be difficult to decide which o f the two causes gives rise 6 It is of course more representativeto measure the amount of annealing in real space, e.g. through f ~b2 (x)dx (see Section 4). For example, if we focused on q = 0 for a conserved field we would not see any annealing. Measuring the relaxation at a finite q is also sufficient to display an interpolation.
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M.F. Zimmer I Physica A 211 (1994) 365-380
0.5 l"
~-~ t
0.3
iI
" ....... ..... .......
~_o
~=.002 ~=.01 ~=.05
0.1
-0.1 0.0
. 0.2
.
. 0.4
. 0.6
0.8
.0
Fig. 3. T/c"f-f -- Tc0 vs. F e for three different waiting times corresponding to a = 0.002, 0.01 and 0.05 (recall
a ~ l/t-). Other parameters are the same as in Fig. 2. Tc(a) -Tco = V2o/ro= 0.6 and Tc(q) -Tco = O. The graphs exhibit how a slower impurity as well as a longer waiting time lead to a lower Tffcff.
0.8 0.6 o 0.4 P
0.2 0.0 -0.2 0.00
0.05
0.10
0.15
0.20
ot
Fig. 4. Tffcff -ZcO VS. a for Ftp = 1, 0.1, 0.01, 0.001, respectively,for the curves from top to bottom. Recall a ,,~ 1/T. Corresponding to the annealed state, there is only one value of Tffcfy as ~ ~ co when the relaxation rate is nonzero. to the effect. The situation is further complicated by the fact that the entanglement o f the gel severely impedes global relaxation, while local relaxation still happens relatively quickly. This could qualitatively change the physical picture described above. On a small scale the entanglements are not so important, and so local relaxation proceeds relatively unhindered. We expect that this local annealing would shift the critical temperature upward, as it did in the Gaussian model. On a larger scale (for some finite time t-) the gel strands would appear to be more immobile, or quenched, and so we would expect a different Tc. I f the critical temperature for the quenched p o l y m e r is lower (as we would expect), then upon lowering the temperature, the phase separation would begin near the p o l y m e r strands first, since they could anneal locally. The temperature would probably have to be lowered beneath the quenched critical temperature before phase separation taking place over the entire system could be observed.
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4. Bdow
To(a)
We cannot go below Tc(a) in the Gaussian model (except for ] A±t [<< 1) without adding higher-order terms for stabilization. However, this makes it extremely difficult to study in a field theoretic way. So in order to proceed further we make a simulation of the system using a cell dynamical model [5]. Such a scheme is known to be an efficient means of simulating statistical systems at a mesoscopic level, as has already been shown with their use in studying spinodal decomposition [ 5 ]. In place of a discretized partial differential equation, the CDS (cell dynamical scheme) exploits a mapping (in time) of spatial cells, which in the limit of small time and space discretizations, reduces to the usual partial differential equations that would be used. In the following case we model a nonconserved host ~p interacting with a conserved impurity ~o. The dynamics are defined by d/t+l
-----
f(d/t) + D ( ((d/t)) - d/t) + v~ot + B~h ,
(4.1)
~Ot+l = (~Ot "~ F~ot [(. 5L"- ~ot) - ((.~-- q)t))] ,
(4.2)
.T" = fl(~p) + D ( ((gt)) - 9t) + vd/t + B,rlW ,
(4.3)
where, for simplicity, f ( d / ) = Ad/ for l A d / [ < 1 and sgn(d/) otherwise, and fl(~o) = A1~o for IA1¢ I< 1/s and s. sgn(q~) otherwise; A and At are positive constants. These functions are the maps previously mentioned, which are constructed so that the dynamics of the ~ and q~ field are nonconserved and conserved, respectively (see Ref. [5] for more details). Random initial conditions are used for d/ and ~0 to represent the high temperature phase. The interaction v plays the same role as in the Gaussian model. The strengths of the white noise terms, ~ and ~7,, are determined by the positive constants B and B~. Local averages are defined via ((d/t)) = 12nnd/t + 1.~nnnd/t; (n)nn denote (next) nearest neighbors. The total amount (or magnitude) of the impurity is limited by s, a constant < 1; the impurity relaxation rate is proportional to F ~ . Not accounting for the shift in Tc due to fluctuations, the value of A corresponding to the bare, or unshifted Tc(q) is Ac(q) = 1. Simulations have shown that when A is smaller than a critical value A c ( a ) 7 , there is no phase transition. Fluctuations have automatically been accounted for in A c ( a ) , as it was determined from a simulation that included noise s . Also, one must not overlook the fact that an increase in temperature is equivalent to a decrease in A, that is Ac(q) > Ac(a), but Tc(q) < Tc(a). Random initial conditions are used for ~p and q~ to represent the high-temperature phase. With the physical parameters fixed (i.e., fixed A), the amount of phase separation" observed depends solely on the waiting time L The evolution of fd/2 dx (i.e., the structure factor S ( k ) at k = 0), which gives a measure of the phase separation, is given in Fig. 5 for a typical run. The graph shows a fairly well defined time at which significant 7 When Ai = 0.95, v = 0.2, s = 0.2, B~ = B = 0.1, D = 0.35, the value of A correspondingto the Tc(a) is Ac(a) = 0.24 4- 0.01. 8 It can be shown that the kinetic coefficients and the parameters in .T" and f may be redefined so that the fluctuation-dissipation theorem is always satisfied.
M.F. Zimmer / Physica A 211 (1994) 365-380
375
8000
6000
%
4000
2000
0 2000
4000
6000
8000
10000
time
Fig. 5. f~p2 dx vs. time for the host when A = 0.7, system of size I 0 0 X 100.
At = 0.95, F~z = 0.1, v0 = 0.2, s = 0.2, B = 0.01 for a
phase separation develops. Figs. 6(a-d) show the spatial pattern of the host field for the same system as in Fig. 5. More interesting is that this pattern closely resembles that of a phase-separating conserved host. This is direct evidence of the enslaving of the host by the impurity. Such a behavior has also been observed in surfactant systems [ 11,14], except there the enslaving was more pronounced since there was no penalty for gradients of the impurity. This led to a pronouced concentration of the impurity on the domain walls, which had the effect of reducing the surface tension in these models. Because the driving force for motion of the interfaces is proportional to the surface tension, this leads to a dramatic reduction in the rate of relaxation. In this CDS model there is no direct link with the bare parameters of the Gaussian model, but analogous conclusions can still be reached. As already described, we obtain the phase diagram by quenching to some temperature (or A), waiting a time L and then measuring f~k2dx. We have the freedom to perform this numerical experiment for a number of impurity relaxation rates, the results of which are given in Fig. 7. Scaling the f~b2 dx curves so that they are of uniform height, we see that lowering F~, shifts the curves to lower temperatures (or higher A). This demonstrates how the effective Tc is dependent upon the relaxation rate of the impurity, decreasing (increasing) with a decrease (increase) in F~s. There is a spread in temperature (or A) over which f~b2 dx changes significantly, and thus we have evidence of how the slow relaxation of the impurity smears the coexistence curve. As one expects, when F ~ --~ 0 (i.e. quenched impurity) there is no significant phase separation until T = Tc(q) (or A = A~(q)). Also, when / ' ~ ~ 1 (i.e. annealed impurity), the host and impurity relaxation rates are comparable, and significant phase separation occurs near Tc(a) (or A = Ac(a)) for large enough L For this CDS model it is possible to find A eff (or equivalently, Te l f ) a s a function of F¢~, with ? fixed. We define Acelf for each of the several curves in Fig. 7 as the value of A at which f~b 2 dx (or more generally, (f~b2dx)~ -- f¢2dx - (fCdx) 2) reaches half its maximum. The resulting dependence of A elf on the impurity relaxation rate is
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M.F. Zimmer / Phys~a A 211 (1994) 365-380
(Q)
(b)
(c)
(d)
Fig. 6. Evolutionof the host field as it phase separates, at time steps (a) 1500, (b) 2500, (c) 3500 and (d) 4500. Parameters are the same as in Fig. 5. in qualitative agreement with that of the Gaussian model, shown in Fig. 3. (f~p2dx)c in the A - ~ plane (see Fig. 8) is given for a range of off-critical quenches in the case of F~, = 0.05, with other parameters being the same as in Fig. 7. It tends to be larger (i.e. the shading is darker) near ~p = 0. All points lie in the two-phase region for the annealed case. The effective Ac, which is where the shading gradually changes, lies somewhere between A¢(q) and A t ( a ) ; exactly where it is depends on the relaxation rate of the impurity and the duration following the quench. Also, the measure of the phase separation (f~p2dx)c changes smoothly about A elf, which is why we say the coexistence curve is smeared. Finally, when F~, was decreased, the time needed to observe phase separation in-
M.F. Zimmer
Physica A 211 (1994) 365-380
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8000 6000 ¢ %,.~4000 2000 0 0.4
0.6
0.8
1.0
A
Fig. 7. f ~ 2 dx vs. A for F~t = 0.25, 0.2, 0.15, 0.1, 0.05 (left to right); to normalize, they were scaled by 1, 1.05, 1.11, 1.25 and 1.797, respectively. Other parameters a r e / = 4 0 0 0 , At = 0.95, v0 =0.2, s = 0.2, B~ = B = 0.01, and the system size is 100 x 100. The data were averaged over four sets of initial conditions. Disregarding fluctuation effects, A = 1 corresponds to T = Tc(q). The value of A corresponding to Tc(a) is Ac(a) = 0.24 + 0.02, and accounts for fluctuations.
creased so as to keep their product approximately constant (provided the separation had not saturated). Since a similar constant for the host was not rescaled, this cannot be claimed to be due to a simple time rescaling. Rather, it demonstrates that the impurity acts as a bottle-neck for the system when its relaxation rate (w0,) is much less than that of the host. In this situation w~t acts as an interpolation parameter.
5. A difficulty in Model C Enough has been learned in the previous section to shed some light on certain problems arising in Model C [6,7,13]. The model was introduced to study the effect of a conserved energy field E on the dynamics of an N-component nonconserved order parameter ¢ near its phase transition point. The equations of motion are
at~ = -Fo
6n[~p, E] 8~0 + v¢,,
atE= AoX726H[O'E] + l'E, 8E with noise v~ and ve chosen so that e -Htq''~j is recovered as the equilibrium distribution. For this model there is instead a coupling ~k2E in the Hamiltonian H: H[~b, e]
= far {l [r0~92
+ ( V ~ b ) 2 ] + uo~//4 -~- "~o~/t2E "['- 1 E 2 -{- soB} •
(5.1)
could be taken to represent a set of spins, and for our case, E could represent a set of (conserved) mobile impurites. This model is known to have renormalization group fixed points at A = 0, c~, and a finite value, where A is the renormalized ho ( ~ Ao/Fo). However, the case A = 0 is considered ambiguous because of a noncommutativity in the
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M.F. Z i m m e r / Physica A 211 (1994) 3 6 5 - 3 8 0
0.651
A
0.99 -0.14
0
0.14
¢ Fi~ 8. A olot of (~b2dx) = f~b2dx - (fCdx) 2 in the A-~b plane for F~t = 0.05; Ac(q) ,~ 1 and Ac(a) ,.~ ().24. D~k~ershading represents larger values. All points m the Figure he m the two phase regime when the impurity is annealed. The curve in Fig. 7 for F~ = 0.05 corresponds to the ~p = 0 line in this Figure. limits ~ --~ 0 and e --* 0 (e = 4 - space dimensionality), and the existence of dynamic scaling has been questioned [7,13]. The main point of this section is merely to suggest (not argue) that this problem could perhaps be symptomatic of a more physical problem that occurs in this situation, which stems from the noncommutativity of the limits A --~ 0 and ? --, c~. That is, there are marked differences between systems with very small and vanishing relaxation rates when the total relaxation time of the system (t-) becomes large. If the relaxation rate of the field is we, the field should appear quenched if toE? << c and annealed if tOE? >> C (where c ,-~ O ( 1 ) constant). In the case of Model C, it is known there is a fixed point at A = 0, which we interpret as A = 0, and not F = c~
M.F. Zimmer / Physica A 211 (1994) 365-380
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(A and F are the renormalized A0 and F0, respectively). This implies ihat at that point the E-field should act as quenched. However, to understand the behavior of the system near this point, the stability of the fixed point must also be considered, which means allowing toe to have some small but finite value. But once toe is allowed to be finite, whether E acts as quenched or annealed becomes dependent on the size of ?. If the initial conditions used in studying the model are set in the infinite past (i.e. ? ~ :x~), then E will act as annealed; yet it is quenched at the fixed point. Of course it is possible for this dependence on ? to show up in calculations done at the fixed point. The calculation is not attempted here, but it is suggested that accounting for initial conditions could possibly act as a remedy for the difficulties encountered with a RG fixed point associated with a vanishing relaxation rate. Initial conditions of a quenched E-field would be natural because then the limit tOE --~ 0 would make sense for all times, particularly ? ~ 0. This could possibly be useful for dealing with some graphs, such as those cited as causing problems because they had the feature they could be separated into two pieces, connected only by E-propagators [13]. In the limit of ? ~ 0 these graphs would disappear, because they would not be allowed due to the quenched initial conditions on E. Ultimately, it seems the size of toE? as toe ~ 0 and ? ~ c~ is central to knowing the behavior of the theory.
6. Concluding remarks Previously, it was not discussed how to describe impurities that did not fall into the standard classification of quenched or annealed. We have given such interpolating correlation functions in the disordered phase of a simple but illustrative model. An effective critical temperature (T~ f f ) was identified, which depended on the relaxation rate of the impurity (F~I) and the duration of the quenching experiment. Te f f smoothly changed from the quenched critical temperature (Tc(q)) (for F~I --~ O) to the annealed critical temperature (Tc(a)), up to a small shift that was dependent on the duration of the experiment. This effective Tc is not associated with a divergent correlation length. Also, the model suggests the coexistence curve is smeared about Telf. These effects should be observable in systems where the relaxation rate (to) is sufficiently small. The key factor is that the experiment should be able to sample the ranges to? << 1, >> 1. If this can be achieved, then it should be possible to compare the results for the evolution of the structure factor, the effective critical temperature, and the coexistence curve. Below T¢(a) (and above To(q)), a field theoretical model is not the most expedient way to understand the system. Using a cell-dynamical scheme, a nonconserved host interacting with a conserved impurity was simulated. Again an effective Tc was identified, about which the coexistence curve was smeared. The nonconserved host was slaved to the impurity, phase-separating with a pattern that was morphologically similar to that of a conserved host. Also, since typically the time dependence of the characteristic length of a phase separating nonconserved (conserved) system scales as t 1/2 (t 1/3) for t ~ c~z, it could be interesting to see what happens to the time dependence in this case.
The importance of initial conditions when a relaxation rate (w) approached zero was pointed out in Model C. The secondary (impurity) field could act as quenched or annealed, depending on the magnitude of wf, where f is the duration of the experiment. To begin with a system at equilibrium, when the impurity has a vanishing relaxation rate, the impurity should be taken as initially quenched.
Acknowledgement
I wish to thank Yoshi Oono for a number of useful suggestions and thoughtful discussions, and for critically reading the manuscript. I would also like to thank Glenn Paquette for discussions, and T. Kawakatsu for bringing some relevant literature to my attention (Refs. [ 11,141). This research was supported by the National Science Foundation through Grant No. DMR 89-20538 administered by the University of Illinois Materials Research Laboratory. References [ 11 K.A. Miiller, in: Lecture Notes in Physics, vol. 104. Charles P. Enz, ed. (SpringwVerlag. 121 P.-G. de Gennes, 1. Phys. Chem. 88 (1984) 6469; K.-Q. Xia and 1.V. Maber, Phys. Rev. A 36 (1987) 2432. [ 31 K.-Q. Xia and I.V. Maber. Phys. Rev. A 37 (1988) 3626. [4] B.I. Halperin and C.M. Varma, Phys. Rev. B 14 (1976) 4030; T. Schneider and E. Stall, Phys. Rev. B 16 (1976) 2220. [5] Y. Oono and A. Shinozaki, Forma 4 (1989) 75; Y. Oono, IEICE Trans. E 74 (1991) 1379; Y. 0x0 and S. Fui, Pbys. Rev. Let. 58 (1987) 836. [6] B.1. Halperin, PC. Hohenberg and S.-K. Ma, Phys. Rev. B 10 (1974) 139. L7] E. Brezin and C. De Dominicis, Phys. Rev. B 12 (1975) 4945: 12 (1975) 4954. [Sl H.K. Janssen, B. Schaub, and B. Schmittmann, 2. Phys. B 73 (1989) 539. [9] C. De Dominicis, Phys. Rev. B 18 (1978) 4913. [IO] A. Shinozaki and Y. Oono, Phys. Rev. E 48 (1994) 2622. [ 111 M. Iaradji, H. Guo, M. Grant and M.J. Zuckermann, J. Phys. A 24 ( 1991) L.629; D.I. Srolovitz and G.N. Hassold, Phys. Rev. B 35 (1987) 6902. [I21 J. Villain, 1. Physique 46 (1985) 1843; D. Andelman and J.-F, Joanny, Phys. Rev. B 32 (1985) 4818; D.S. Fisher, Phys. Rev. Lett. 56 (1986) 419; D.A. Huse. Phys. Rev. B 36 (1987) 5383. [ 131 B.I. Halperin, PC. Hohenberg and S.-K. Ma, Phys. Rev. B 13 (1976) 4119. [ 141 T. Kawakatsu and K. Kawasaki, J. Colloid Interface Sci. 145 (1991) 420. [ 151 PC Martin, E.D. Siggia and H.H. Rose, Phys. Rev. A 8 (1973) 423; H.K. Janssen, Z Physik B 23 (1976) 377; R. Bausch, H.K. Janssen and H. Wagner, Z. Physik B 24 (1976) 113.
1978)
Physica A 211 (1994) 365-380
Interpolating between quenched and annealed disorder Michael E Zimmer 1 Department of Physics, Materials Research Laboratory, and Beckman Institute, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, 1L 61801-3080, USA
Received 30 August 1993; revised 1 July 1994
Abstract A host-impurity system is considered, and the effect on the host due to a slow relaxation of the impurity is studied, when each can be described by a continuous field. Langevin dynamics are assumed for each field, along with a special choice of initial conditions that corresponds to a quenched equilibrium. Equal-time correlation functions interpolate between the limiting cases of a quenched and annealed impurity. An effective critical temperature may be defined, which is dependent on the relaxation rate of the impurity and the duration of the experiment. A simulation for the case of a conserved impurity and nonconserved host is used to demonstrate how a coexistence curve can become smeared when the relaxation rate of the impurity is small. The impurity enslaves the host, causing it to phase separate with a pattern that is more reminiscent of a conserved field. Also, it is suggested that the approach used could shed some light on the difficulties with one of the fixed points of Model C.
1. Introduction Impurities are typically classified as either quenched (i.e. relatively immobile) or annealed (i.e. relatively free to move about). While this is useful for describing most cases, it does not account for those impurities that can only partially anneal on the time scales o f the experiment. A possible example could be realized in SrTiO3, where slowly relaxing defects are considered to be crucial for an understanding o f the central peak in its displacive structural phase transition [ 1 ]. Also, a gel immersed in a binary fluid has been viewed as a quenched random impurity problem [2]. However, in such I Current address: IFF, Forschungszentrum Jiilich, D-52425 Jiilich, Germany. 0378-4371/94/$07.00 (~) 1994 Elsevier Science B.V. All fights reserved SSD10378-4371 (94)00178-2
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a system the polymers (impurity) are free to move around to some degree, and it has been wondered how to quantify the effect of the slow relaxation of the polymers on the phase separation of the fluid [3]. In addition, there have been studies [4] treating the slow annealing of impurities, e.g. the annealing of the Cr 5+ impurities substituted for As in KD2AsO4. However, the long-time annealing of the Cr 5+ was not considered and so any discussion of an interpolation becomes precluded. Besides these, other works treating annealed impurities tended to concentrate on the asymptotic behavior, rather than on any kind of a connection between the two equilibrium averages. These studies dealt with surfactants and polymer blends; their similarities will be pointed out in turn. In this paper we shall investigate the effect of a slow impurity, first by studying a simple model above the (equilibrium) annealed critical temperature (Tc(a)), and then by simulating a similar model below To(a). Above Tc(a), we give correlation functions that interpolate between the quenched and annealed cases. This is done for the simple but illustrative case of a Gaussian model; it is sufficient to reveal the qualitative behavior we seek (and we believe that adding higher order terms would only complicate the model without adding any new physics). We can define an effective critical temperature, Teff, that ranges between the (equilibrium) quenched critical temperature (T~(q)) and Tc(a), being closer to Tc(q) when the relaxation rate of the impurity,/'~, is smaller. Representative plots of Teyf vs. F~ and Teff vs. duration of experiment, ~', are given for comparison with numerical experiment. Below the annealed critical temperature it is more useful to proceed via simulation, as direct analytical approaches are less effective. Using a cell dynamical scheme [5] we simulate, as an example, a nonconserved host and a conserved impurity below the annealed critical temperature. A smearing of the coexistence curve about an effective critical temperature is demonstrated. In this case the host is enslaved by the impurity, and the phase separation is morphologically similar to that of a conserved ho.~t. In some models there arises the case where the relaxation rate of one of the fields vanishes; such is the case at one of the renormalization group fixed points of Model C [6,7]. In studying the stability of these fixed points, small but finite relaxation rates must be allowed for. However, this is seen to have the effect of making the field act annealed, when it was originally quenched at the fixed point. This is mentioned as being a possible reason for the ambiguities arising with this fixed point for this model. In Section 2 the statistics of the limiting cases of quenched and annealed impurities are reviewed. A dynamical interpolation is argued for and the necessary experimental scenario discussed. In Section 3 dynamics are introduced via Langevin equations in order to calculate an explicit interpolation on a model bilinear Hamiltonian. Among other things, graphical dependences are found for an effective critical temperature vs. the impurity relaxation rate, as well as the total time of relaxation. In Section 4 the particular case of a nonconserved host interacting with a conserved impurity is simulated below its annealed critical temperature. The evolution of ¢2 (~p = host field) integrated over the system is studied as a function of temperature and duration of quench. It is meant to provide a measure of the phase separation and is in qualitative agreement with the results of the bilinear model. In Section 5 the relevance of the simultaneous limits
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to ~ 0, ? ~ cx~ to renormalization group fixed points at to = 0 is discussed (to = relaxation rate, ? = duration of relaxation). Finally, concluding remarks are made in Section 6.
2. General considerations To make progress using the standard techniques, it is necessary to assume a Hamiltonian High, ~o], which determines the joint probability distribution of the host ((b) and the impurity (~o). As this Hamiltonian is supposed to determine the static behavior, it holds equally true in the case when the dynamics of the impurity are frozen. Thus, for H[~b,~o] polynomial in its arguments, we define HI,o] -- H[0,~o] (which represents self-~p interactions) and set H[~p,~o] = n[~bl~o] + H[~o];
(2.1)
H[~bl~o] is the Hamiltonian for interactions of ~0 given a fixed distribution of ~o. It must not be forgotten that this holds true at only a bare (unrenormalized) level, since a quenched and annealed Hamiltonian will in general be renormalized differently. The above three H ' s are all that is needed to calculate correlation functions in the two equilibrium limits. First, in the quenched limit, when the host has relaxed about a frozen impurity, the equilibrium correlations of ~b may be determined from the free energy ( F ) :
F=-fT~ozP[~oi]ln{fT~Oe-m*l~'l},
(2.2)
where P [ ~oI] is the probability distribution for ~'1, and D o and D~Ol represent integrations over all field configurations, as usual. Were the impurity instead able to explore the host over some time scale (as set by experiment), then we would rather calculate the free energy, using
F = - In {fZ~0~9,,
e-O~C"*t~}.
(2.3)
Given that the above two expressions for F correspond to different limits of the impurity mobility, one may ask whether it is possible to form an interpolation between them. While there are a number of artificial interpolations that may be invented, a physical one presents itself. The scenario is to take the initial condition of ~p relaxed about a quenched ~o, and then to use dynamics that evolve ~p and ~o towards a mutual (annealed) equilibrium. Away from criticality, where relaxations typically behave as e -°'r (w = relaxation rate, ?= duration of relaxation), it is clear that the quenched and annealed (static) averages are recovered (from the equal-time correlation function) in the limits w? ~ 0 and w~"---, c~, respectively. Finally, because to is normally fixed by other parameters, we may just consider the time ? as the interpolation parameter. On a mathematical level it is clear that the above provides for an interpolation. In applying this to real systems however, our requisite initial condition forces us to focus
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only on cases where the impurity relaxes much more slowly than the host. That is, in preparing the system with a temperature jump, we require that after some time the host is effectively annealed and the impurity still effectively frozen. To assure this comfortably would require at least (roughly) two orders of magnitude difference in the relaxation rates of ~p and ~p. A forced application of the formulas would necessitate "holding down" the impurity until the host relaxed, and then letting go at the initial time. Finally, we note that the difficulties in applying these equations in intermediate cases mirrors the difficulty in choosing how to define an average of an evolving system.
3. Disordered phase In this Section we discuss for a particular example the interpolation between quenched and annealed correlation functions above T¢ (a), the annealed critical temperature. If the dynamics of the host and impurity are known, it is always possible to make such an interpolation, at least in principle. In the disordered phase and away from the quenched and annealed critical temperatures, a Gaussian model suffices to reveal the main qualitative features of an interpolation. Higher order terms could be included, but at this stage they would only complicate the picture. This model could be relevant in the disordered phase for those defects which, upon fluctuating, locally break the symmetry of the host. The defects are assumed to be able to (at least eventually) explore their entire phase space (i.e. ergodicity is not broken). For the sake of example we consider the case
U[~,, ~,1 = f d a x
{~ [r0~ 2 + ( V ~ ) 2] + ½ [r0~,~ + ( V ~ , ) ~] + u O ~ l } .
(3.1)
There are energy penalties for self-interactions and gradients of each field; there is also an interaction term between the two fields. 7-0, ro and v0 are positive constants. By inspection, the correlation for ~b-fluctuations is (~l,q = f eiqx~b(x)dx) (~Jq~.l_q)
=
(,)(~ 1 - u2 )(tp) - 1
,
(3.2)
showing annealed critical temperature Tc ( a ) = Tw + v2/7-0; we define X g, = ( ro + q2) -1 and X~ = ( t o + q 2 ) -1- For vo > 0 ( < 0), a positive fluctuation in ~b will favor a negative (positive) fluctuation in ~t, which in turn will favor a positive fluctuation in ~,. So regardless of the sign of vo, the interaction enhances correlations in ~b, as well as ~Pt (this follows from the symmetry of the two fields in Eq. (3.1)). For the impurity (with q~tq = f eiqxq~l (X) ), (~Olqq~l_q) = ( X ; 1 -- U2X,) -1 , f r o m w h i c h w e m a y identify an effective 7-, 7-eff = 7-0 _ U2/FO. AS r 0 ~
(3.3) O, r e f f m a y
become negative, indicating a phase transition. This "annealed" phase transition happens
M.F. Zimmer / Physica A 211 (1994) 365-380
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at the same temperature for both fields, when ro~'0 = v~. Of course, this model is unstable in this case because there are no higher order stabilizing terms (such as ~4 or q~4). In the case when the impurity is quenched, the appropriate Hamiltonian for the host ~ (for a given impurity configuration q~t) would be the same as Eq. (3.1) except without the quadratic terms in q~t. In the quenched case, because the interaction is bilinear, there is a change in ( ~ ) only because of a change in (~); it is easily seen 2 2 that (g/q¢_q) = X¢, + OoX~X¢,. Because of its quenched nature, correlations of ~oi will not change due to interactions with ~. The self-q~t interactions in F_x1. (3.1) immediately give (~Olq~Ot--q) .~ X~o" For the purpose of describing the interpolation, any dynamics may be used, but here we choose the following Langevin equations out of simplicity and also because they are useful in many physical models. In this example the equations are for nonconserved fields (i.e. f d d x all(x, t) is allowed to change; the simplest way to model conserved fields is to change F --~ F q 2) 0e~, = - F ~
0,~Ol = - F ~ ,
6 H [ ~9, ~Ol] 8~0 + v,
(3.4)
8 H [ ¢ , ~ot ] 6~Ol + ('
(3.5)
with zero-mean Gaussian white noise (~,( x, t ) u( x', t') ) = 2 F c B ( x - x ' ) 8 ( t - t'),
(3.6)
and similarly for (; ( ) denotes a noise average. There are no correlations between ~, and (. The initial conditions ¢ ( x , 0 ) -= Ct0(x), ~oi(x,O) - q~lo(x) are set with the weight e -H[O°l~'°] , where Mt
ol
,o] =
fd"x(
[ro~b ' d2 + (Vrpo) 2] +
.
(3.7)
(the meaning of the primed constants is explained below). This is the interaction energy for the ~9 field for a given impurity field gOlO (at t = 0). As it is quenched at t = 0, the impurity must be averaged over with the probability distribution at that time, which is denoted by P [ (Pl0] : P[goi0] o( exp
-
dax [TOq~;O+ (Vq~,o) 2]
)
=
.
(3.8)
The form for the above two equations is determined from H[~O, q~t ], the Hamiltonian for interactions of impurity and host, which gives the energy of self-interactions between fields. It is appropriate to consider it as being composed of a piece with only selfinteractions of q~l, H[q~l], and the piece H[O I ~ot], which was already defined. That is, at t = 0 we set (as previously mentioned) H[O,q~t] = H[¢o I ~OlO] + H[~OlO].
(3.9)
That we can write such an expression at the bare (unrenormalized) level is obvious. However, when higher-order interactions are present, corresponding parameters on the
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right and left hand sides of this equation will be affected differently; quenched and annealed interactions renormalize parameters in different ways. It is for this reason that the parameters which appear in Eqs. (3.7) and (3.8) even though are primed, they originally represented the same physical interactions as in Eq. (3.1). This is done merely as a bookkeeping device. Also, it is possible to write this in terms of a generating functional, which serves as a basis from which one can discuss its renormalizability, in a field-theoretic fashion 2. A simple calculation yields the following cumulants, for T > T¢(a) 3
FO (~bq(t)~b_q(t))= ( a + - - ' l _ )
[ A+ - r°O A_ (1
Xo¢
(q~lq(t)q~l_q(t)) c -
e-2a_t)
-
A- - ~aO A+
(1 - e
e_a+t
F¢/ (A+~-A_)
[A+-~o¢, ~-+
(1 -
-2a+t)
]
2, (3.14)
e_2~+t)
A_ - o)o ~ A_
(1 - e - 2 a - t )
~oo)--__ ("-~-? a _ - a, o) [e -'~+t - e-A-t] 2, -XO0 (a+ --'~+
]
(3.15)
where the effective relaxation rates are a± = ½
+
i ½
+44r ,r ,
-
(3.16)
and oJ0 = F~(ro + q2) and oJ¢, = F¢,(ro + q2) [14]. This correlation function is for a nonconserved host and a nonconserved impurity. X0~ comes from Eq. (3.5) and is 2 The generating functional for our problem is
W[J, K]
= [
J
Y)~otP[cp/0] In W[J, K
I ~,0]
(3.10)
9
W[J, Kl,,o]=/79¢7~,790oe-nt*ol~'o~6[O-O~lSt~1-,,clexp(fdaxdt(JO+K,l)). (3.11) One can argue that there will not be any new divergences induced by this formalism. This is aside from the short time divergences that result from when one of the external points of certain correlation functions approaches the initial time surface. The parameters and fields that are restricted to the initial time surface will be renormalized as in the corresponding quenched problem. The renormalizations of parameters off the surface seems to be the same as in the bulk theory (i.e. the annealed case). For a discussion of this see [8]. 3 The usual correlation functions are given by: u2 ~ v2
(#) = ( # ) c
+
oXu~oa¢
(~-
~-2-)2
[A_e-a+ ' - A + e - ~ - ' ] 2
o~X~ r, [a_(x+ - ~oo)e-;t+t (~,~1) = (¢,~,)c + (,~+X--r--,2
(3.12)
A+(A_ - o ¢ , ) e - a - ' ]~2 .
(3.13)
M.F. Zimmer / Physica A 211 (1994) 365-380 20
20
15
15
-.~1o
10
371
V
5 0
5 I
i
-I .0
-0.5
I
0.0
0.5
.0
0 -I .0
-0.5
0.0
0.5
1.0
q
Fig. 1. (~.np)c is plotted with the values r0 = 0.15, ~'0 = 1, v2 = 0.1, re = 4., and F,t = 0.1 for: (a) nonconserved host, nonconserved impurity at times 0, 10, 100 ; (b) conserved host (i.e. F¢ ---} Fcq2), nonconserved impurity at times 0, 100, 1000, I0000. As time evolves the plots move upwards. equal to (r~ + q2) -1. However, since ro and r~ represent the same physical interaction,
Xo¢ = (ro + q2) -1. For T < Tc(a) this can only be used for short times, since this model does not have the necessary higher order stabilizing terms. It is easy to check that in the limits t ---} 0 and t ---* c~, the correlation functions arising from a quenched and annealed impurity are recovered, respectively. Finally, in the case o f F , I = 0, the impurity is quenched for all time, and it is straightforward to show that this formulation reduces down to the quenched impurity case studied by De Dominicis [9]. For T > Tc(a) the interpolating function (~kq(t)~_q(t)) c is plotted in Fig. 1 for the two cases o f a conserved (to~ = Fcq2(ro + q2)) and nonconserved (toe = F¢(ro + q2)) host with a nonconserved (¢o, = F~,(~'o + q2)) impurity. The graphs depict the evolution o f (~kq(t)~l_q(t)) c from t = 0, where the impurity distribution is quenched, to values large enough that the impurity distribution is effectively annealed. In Fig. l b the host is conserved, and thus (~q(t)~k_q(t)) c cannot change at q = 0. The global conservation leads to a selected momentum scale 4 which approaches zero as time increases. A t larger times it will most likely be difficult to observe since scattering experiments involve only a small subvolume (i.e. an open system), which are not constrained by conservation laws. Such a narrowing peak is reminiscent o f the structure factor in spinodal decomposition [ 10], where it reflects the dominant role played by the correlation length in bringing about dynamical scaling. In studies involving an annealing impurity (none o f which involved energy penalties for impurity gradients) this peak would saturate [ 11 ], reflecting the maximum size the correlation length could grow to on experimental time scales. Finally, the other two combinations, conserved impurity and conserved or nonconserved host, lead to interpolating functions that are qualitatively very similar to Fig. l b and so are not included 5 . 4 This length scale was found not to scale for times up to ,-, 10 9, with all constants ,~ O( 1). 5 The fact that there is a "pinning" at q = 0 when the host is nonconserved and the impurity is conserved can be shown to be due to the Gaussian nature of the theory (i.e., higher order terms eliminate this).
372
M.F. Zimmer / Physica A 211 (1994) 365-380 0.1
,
,.<' - 0 . 2
.'" -"
..'/
- ....
•
P~,=O P~=.01
P~=.l
P~= 1
-0.5 -0.1
0.2
0.5
0.8
%
Fig. 2. The effective relaxation rate A- for a nonconserved host and impurity is plotted against ro = T - Tco for F¢ = 1, v2 = 0.6, ro = 1, q = 0 and F , = (i) 0, (ii) 0.01, (iii) 0.1 and (iv) 1. The intersection of -or with the A curves occurs at lffcff - Tco, by definition. The dotted line is - a . If f is defined as the duration of the experiment, then above Tc(a) quenched and annealed behavior will be observed when A_3 << c and A_3 >> c, respectively (assuming A+t >> c, where c is an O ( 1 ) constant). This defintion will necessarily include a qdependence 6 ; here, q = 0 is chosen. Below Tc(a) phase separation can only be observed if there has been sufficient (exponential) growth. Above Tc(q), the quenched critical temperature, this amounts to - a _ 3 > c, since only A_ can be negative here. So given 3, phase separation will only be observed if - A _ > c/? =_ or. In Fig. 2 we plot A_ for several values o f F~,, which intersect a representative - a line. By definition, the intersections take place at r0 = Te l f - Tco, which in turn leads to the dependence o f the effective Tc on the relaxation rate o f the impurity (see Fig. 3). As defined, Te l f ranges from Tc(q) to T~(a), up to a small shift that is due to a finite 3; thus we confine our attention to large L This effective Tc correctly changes from ~ T~(q) (for F,o,/F¢, ~ O) over to ~ Tc(a) (for F ~ J F ¢ ~ O ( 1 ) ) , and tends to be closer to To(a) for larger L If 3 --~ cx~ (i.e. at equilibrium) then Teyf would equal Tc(a) everywhere, except where F~, = 0, as must be the case. The sensitivity o f T[ y / o n the time 3 is demonstrated in Fig. 4. It gives a quantitative measure of how the absolute value o f the effective Tc's change, as well their values relative to other T [ f f ' s at a given relaxation rate. If in a series of experiments we critically quenched to several temperatures between Tc(a) and Tc(q) and waited a time L we would have several values o f A_3 and would thus see different stages of development of the phase separation. In this sense the coexistence curve is smeared over a range of temperature. This type of effect has been observed near the phase transition point of binary fluids in gels [3]. However, one can also argue [ 12] that the impurity field induces long lived metastable states in the host field. In this case it may be difficult to decide which o f the two causes gives rise 6 It is of course more representativeto measure the amount of annealing in real space, e.g. through f ~b2 (x)dx (see Section 4). For example, if we focused on q = 0 for a conserved field we would not see any annealing. Measuring the relaxation at a finite q is also sufficient to display an interpolation.
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0.5 l"
~-~ t
0.3
iI
" ....... ..... .......
~_o
~=.002 ~=.01 ~=.05
0.1
-0.1 0.0
. 0.2
.
. 0.4
. 0.6
0.8
.0
Fig. 3. T/c"f-f -- Tc0 vs. F e for three different waiting times corresponding to a = 0.002, 0.01 and 0.05 (recall
a ~ l/t-). Other parameters are the same as in Fig. 2. Tc(a) -Tco = V2o/ro= 0.6 and Tc(q) -Tco = O. The graphs exhibit how a slower impurity as well as a longer waiting time lead to a lower Tffcff.
0.8 0.6 o 0.4 P
0.2 0.0 -0.2 0.00
0.05
0.10
0.15
0.20
ot
Fig. 4. Tffcff -ZcO VS. a for Ftp = 1, 0.1, 0.01, 0.001, respectively,for the curves from top to bottom. Recall a ,,~ 1/T. Corresponding to the annealed state, there is only one value of Tffcfy as ~ ~ co when the relaxation rate is nonzero. to the effect. The situation is further complicated by the fact that the entanglement o f the gel severely impedes global relaxation, while local relaxation still happens relatively quickly. This could qualitatively change the physical picture described above. On a small scale the entanglements are not so important, and so local relaxation proceeds relatively unhindered. We expect that this local annealing would shift the critical temperature upward, as it did in the Gaussian model. On a larger scale (for some finite time t-) the gel strands would appear to be more immobile, or quenched, and so we would expect a different Tc. I f the critical temperature for the quenched p o l y m e r is lower (as we would expect), then upon lowering the temperature, the phase separation would begin near the p o l y m e r strands first, since they could anneal locally. The temperature would probably have to be lowered beneath the quenched critical temperature before phase separation taking place over the entire system could be observed.
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4. Bdow
To(a)
We cannot go below Tc(a) in the Gaussian model (except for ] A±t [<< 1) without adding higher-order terms for stabilization. However, this makes it extremely difficult to study in a field theoretic way. So in order to proceed further we make a simulation of the system using a cell dynamical model [5]. Such a scheme is known to be an efficient means of simulating statistical systems at a mesoscopic level, as has already been shown with their use in studying spinodal decomposition [ 5 ]. In place of a discretized partial differential equation, the CDS (cell dynamical scheme) exploits a mapping (in time) of spatial cells, which in the limit of small time and space discretizations, reduces to the usual partial differential equations that would be used. In the following case we model a nonconserved host ~p interacting with a conserved impurity ~o. The dynamics are defined by d/t+l
-----
f(d/t) + D ( ((d/t)) - d/t) + v~ot + B~h ,
(4.1)
~Ot+l = (~Ot "~ F~ot [(. 5L"- ~ot) - ((.~-- q)t))] ,
(4.2)
.T" = fl(~p) + D ( ((gt)) - 9t) + vd/t + B,rlW ,
(4.3)
where, for simplicity, f ( d / ) = Ad/ for l A d / [ < 1 and sgn(d/) otherwise, and fl(~o) = A1~o for IA1¢ I< 1/s and s. sgn(q~) otherwise; A and At are positive constants. These functions are the maps previously mentioned, which are constructed so that the dynamics of the ~ and q~ field are nonconserved and conserved, respectively (see Ref. [5] for more details). Random initial conditions are used for d/ and ~0 to represent the high temperature phase. The interaction v plays the same role as in the Gaussian model. The strengths of the white noise terms, ~ and ~7,, are determined by the positive constants B and B~. Local averages are defined via ((d/t)) = 12nnd/t + 1.~nnnd/t; (n)nn denote (next) nearest neighbors. The total amount (or magnitude) of the impurity is limited by s, a constant < 1; the impurity relaxation rate is proportional to F ~ . Not accounting for the shift in Tc due to fluctuations, the value of A corresponding to the bare, or unshifted Tc(q) is Ac(q) = 1. Simulations have shown that when A is smaller than a critical value A c ( a ) 7 , there is no phase transition. Fluctuations have automatically been accounted for in A c ( a ) , as it was determined from a simulation that included noise s . Also, one must not overlook the fact that an increase in temperature is equivalent to a decrease in A, that is Ac(q) > Ac(a), but Tc(q) < Tc(a). Random initial conditions are used for ~p and q~ to represent the high-temperature phase. With the physical parameters fixed (i.e., fixed A), the amount of phase separation" observed depends solely on the waiting time L The evolution of fd/2 dx (i.e., the structure factor S ( k ) at k = 0), which gives a measure of the phase separation, is given in Fig. 5 for a typical run. The graph shows a fairly well defined time at which significant 7 When Ai = 0.95, v = 0.2, s = 0.2, B~ = B = 0.1, D = 0.35, the value of A correspondingto the Tc(a) is Ac(a) = 0.24 4- 0.01. 8 It can be shown that the kinetic coefficients and the parameters in .T" and f may be redefined so that the fluctuation-dissipation theorem is always satisfied.
M.F. Zimmer / Physica A 211 (1994) 365-380
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8000
6000
%
4000
2000
0 2000
4000
6000
8000
10000
time
Fig. 5. f~p2 dx vs. time for the host when A = 0.7, system of size I 0 0 X 100.
At = 0.95, F~z = 0.1, v0 = 0.2, s = 0.2, B = 0.01 for a
phase separation develops. Figs. 6(a-d) show the spatial pattern of the host field for the same system as in Fig. 5. More interesting is that this pattern closely resembles that of a phase-separating conserved host. This is direct evidence of the enslaving of the host by the impurity. Such a behavior has also been observed in surfactant systems [ 11,14], except there the enslaving was more pronounced since there was no penalty for gradients of the impurity. This led to a pronouced concentration of the impurity on the domain walls, which had the effect of reducing the surface tension in these models. Because the driving force for motion of the interfaces is proportional to the surface tension, this leads to a dramatic reduction in the rate of relaxation. In this CDS model there is no direct link with the bare parameters of the Gaussian model, but analogous conclusions can still be reached. As already described, we obtain the phase diagram by quenching to some temperature (or A), waiting a time L and then measuring f~k2dx. We have the freedom to perform this numerical experiment for a number of impurity relaxation rates, the results of which are given in Fig. 7. Scaling the f~b2 dx curves so that they are of uniform height, we see that lowering F~, shifts the curves to lower temperatures (or higher A). This demonstrates how the effective Tc is dependent upon the relaxation rate of the impurity, decreasing (increasing) with a decrease (increase) in F~s. There is a spread in temperature (or A) over which f~b2 dx changes significantly, and thus we have evidence of how the slow relaxation of the impurity smears the coexistence curve. As one expects, when F ~ --~ 0 (i.e. quenched impurity) there is no significant phase separation until T = Tc(q) (or A = A~(q)). Also, when / ' ~ ~ 1 (i.e. annealed impurity), the host and impurity relaxation rates are comparable, and significant phase separation occurs near Tc(a) (or A = Ac(a)) for large enough L For this CDS model it is possible to find A eff (or equivalently, Te l f ) a s a function of F¢~, with ? fixed. We define Acelf for each of the several curves in Fig. 7 as the value of A at which f~b 2 dx (or more generally, (f~b2dx)~ -- f¢2dx - (fCdx) 2) reaches half its maximum. The resulting dependence of A elf on the impurity relaxation rate is
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M.F. Zimmer / Phys~a A 211 (1994) 365-380
(Q)
(b)
(c)
(d)
Fig. 6. Evolutionof the host field as it phase separates, at time steps (a) 1500, (b) 2500, (c) 3500 and (d) 4500. Parameters are the same as in Fig. 5. in qualitative agreement with that of the Gaussian model, shown in Fig. 3. (f~p2dx)c in the A - ~ plane (see Fig. 8) is given for a range of off-critical quenches in the case of F~, = 0.05, with other parameters being the same as in Fig. 7. It tends to be larger (i.e. the shading is darker) near ~p = 0. All points lie in the two-phase region for the annealed case. The effective Ac, which is where the shading gradually changes, lies somewhere between A¢(q) and A t ( a ) ; exactly where it is depends on the relaxation rate of the impurity and the duration following the quench. Also, the measure of the phase separation (f~p2dx)c changes smoothly about A elf, which is why we say the coexistence curve is smeared. Finally, when F~, was decreased, the time needed to observe phase separation in-
M.F. Zimmer
Physica A 211 (1994) 365-380
377
8000 6000 ¢ %,.~4000 2000 0 0.4
0.6
0.8
1.0
A
Fig. 7. f ~ 2 dx vs. A for F~t = 0.25, 0.2, 0.15, 0.1, 0.05 (left to right); to normalize, they were scaled by 1, 1.05, 1.11, 1.25 and 1.797, respectively. Other parameters a r e / = 4 0 0 0 , At = 0.95, v0 =0.2, s = 0.2, B~ = B = 0.01, and the system size is 100 x 100. The data were averaged over four sets of initial conditions. Disregarding fluctuation effects, A = 1 corresponds to T = Tc(q). The value of A corresponding to Tc(a) is Ac(a) = 0.24 + 0.02, and accounts for fluctuations.
creased so as to keep their product approximately constant (provided the separation had not saturated). Since a similar constant for the host was not rescaled, this cannot be claimed to be due to a simple time rescaling. Rather, it demonstrates that the impurity acts as a bottle-neck for the system when its relaxation rate (w0,) is much less than that of the host. In this situation w~t acts as an interpolation parameter.
5. A difficulty in Model C Enough has been learned in the previous section to shed some light on certain problems arising in Model C [6,7,13]. The model was introduced to study the effect of a conserved energy field E on the dynamics of an N-component nonconserved order parameter ¢ near its phase transition point. The equations of motion are
at~ = -Fo
6n[~p, E] 8~0 + v¢,,
atE= AoX726H[O'E] + l'E, 8E with noise v~ and ve chosen so that e -Htq''~j is recovered as the equilibrium distribution. For this model there is instead a coupling ~k2E in the Hamiltonian H: H[~b, e]
= far {l [r0~92
+ ( V ~ b ) 2 ] + uo~//4 -~- "~o~/t2E "['- 1 E 2 -{- soB} •
(5.1)
could be taken to represent a set of spins, and for our case, E could represent a set of (conserved) mobile impurites. This model is known to have renormalization group fixed points at A = 0, c~, and a finite value, where A is the renormalized ho ( ~ Ao/Fo). However, the case A = 0 is considered ambiguous because of a noncommutativity in the
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M.F. Z i m m e r / Physica A 211 (1994) 3 6 5 - 3 8 0
0.651
A
0.99 -0.14
0
0.14
¢ Fi~ 8. A olot of (~b2dx) = f~b2dx - (fCdx) 2 in the A-~b plane for F~t = 0.05; Ac(q) ,~ 1 and Ac(a) ,.~ ().24. D~k~ershading represents larger values. All points m the Figure he m the two phase regime when the impurity is annealed. The curve in Fig. 7 for F~ = 0.05 corresponds to the ~p = 0 line in this Figure. limits ~ --~ 0 and e --* 0 (e = 4 - space dimensionality), and the existence of dynamic scaling has been questioned [7,13]. The main point of this section is merely to suggest (not argue) that this problem could perhaps be symptomatic of a more physical problem that occurs in this situation, which stems from the noncommutativity of the limits A --~ 0 and ? --, c~. That is, there are marked differences between systems with very small and vanishing relaxation rates when the total relaxation time of the system (t-) becomes large. If the relaxation rate of the field is we, the field should appear quenched if toE? << c and annealed if tOE? >> C (where c ,-~ O ( 1 ) constant). In the case of Model C, it is known there is a fixed point at A = 0, which we interpret as A = 0, and not F = c~
M.F. Zimmer / Physica A 211 (1994) 365-380
379
(A and F are the renormalized A0 and F0, respectively). This implies ihat at that point the E-field should act as quenched. However, to understand the behavior of the system near this point, the stability of the fixed point must also be considered, which means allowing toe to have some small but finite value. But once toe is allowed to be finite, whether E acts as quenched or annealed becomes dependent on the size of ?. If the initial conditions used in studying the model are set in the infinite past (i.e. ? ~ :x~), then E will act as annealed; yet it is quenched at the fixed point. Of course it is possible for this dependence on ? to show up in calculations done at the fixed point. The calculation is not attempted here, but it is suggested that accounting for initial conditions could possibly act as a remedy for the difficulties encountered with a RG fixed point associated with a vanishing relaxation rate. Initial conditions of a quenched E-field would be natural because then the limit tOE --~ 0 would make sense for all times, particularly ? ~ 0. This could possibly be useful for dealing with some graphs, such as those cited as causing problems because they had the feature they could be separated into two pieces, connected only by E-propagators [13]. In the limit of ? ~ 0 these graphs would disappear, because they would not be allowed due to the quenched initial conditions on E. Ultimately, it seems the size of toE? as toe ~ 0 and ? ~ c~ is central to knowing the behavior of the theory.
6. Concluding remarks Previously, it was not discussed how to describe impurities that did not fall into the standard classification of quenched or annealed. We have given such interpolating correlation functions in the disordered phase of a simple but illustrative model. An effective critical temperature (T~ f f ) was identified, which depended on the relaxation rate of the impurity (F~I) and the duration of the quenching experiment. Te f f smoothly changed from the quenched critical temperature (Tc(q)) (for F~I --~ O) to the annealed critical temperature (Tc(a)), up to a small shift that was dependent on the duration of the experiment. This effective Tc is not associated with a divergent correlation length. Also, the model suggests the coexistence curve is smeared about Telf. These effects should be observable in systems where the relaxation rate (to) is sufficiently small. The key factor is that the experiment should be able to sample the ranges to? << 1, >> 1. If this can be achieved, then it should be possible to compare the results for the evolution of the structure factor, the effective critical temperature, and the coexistence curve. Below T¢(a) (and above To(q)), a field theoretical model is not the most expedient way to understand the system. Using a cell-dynamical scheme, a nonconserved host interacting with a conserved impurity was simulated. Again an effective Tc was identified, about which the coexistence curve was smeared. The nonconserved host was slaved to the impurity, phase-separating with a pattern that was morphologically similar to that of a conserved host. Also, since typically the time dependence of the characteristic length of a phase separating nonconserved (conserved) system scales as t 1/2 (t 1/3) for t ~ c~z, it could be interesting to see what happens to the time dependence in this case.
The importance of initial conditions when a relaxation rate (w) approached zero was pointed out in Model C. The secondary (impurity) field could act as quenched or annealed, depending on the magnitude of wf, where f is the duration of the experiment. To begin with a system at equilibrium, when the impurity has a vanishing relaxation rate, the impurity should be taken as initially quenched.
Acknowledgement
I wish to thank Yoshi Oono for a number of useful suggestions and thoughtful discussions, and for critically reading the manuscript. I would also like to thank Glenn Paquette for discussions, and T. Kawakatsu for bringing some relevant literature to my attention (Refs. [ 11,141). This research was supported by the National Science Foundation through Grant No. DMR 89-20538 administered by the University of Illinois Materials Research Laboratory. References [ 11 K.A. Miiller, in: Lecture Notes in Physics, vol. 104. Charles P. Enz, ed. (SpringwVerlag. 121 P.-G. de Gennes, 1. Phys. Chem. 88 (1984) 6469; K.-Q. Xia and 1.V. Maber, Phys. Rev. A 36 (1987) 2432. [ 31 K.-Q. Xia and I.V. Maber. Phys. Rev. A 37 (1988) 3626. [4] B.I. Halperin and C.M. Varma, Phys. Rev. B 14 (1976) 4030; T. Schneider and E. Stall, Phys. Rev. B 16 (1976) 2220. [5] Y. Oono and A. Shinozaki, Forma 4 (1989) 75; Y. Oono, IEICE Trans. E 74 (1991) 1379; Y. 0x0 and S. Fui, Pbys. Rev. Let. 58 (1987) 836. [6] B.1. Halperin, PC. Hohenberg and S.-K. Ma, Phys. Rev. B 10 (1974) 139. L7] E. Brezin and C. De Dominicis, Phys. Rev. B 12 (1975) 4945: 12 (1975) 4954. [Sl H.K. Janssen, B. Schaub, and B. Schmittmann, 2. Phys. B 73 (1989) 539. [9] C. De Dominicis, Phys. Rev. B 18 (1978) 4913. [IO] A. Shinozaki and Y. Oono, Phys. Rev. E 48 (1994) 2622. [ 111 M. Iaradji, H. Guo, M. Grant and M.J. Zuckermann, J. Phys. A 24 ( 1991) L.629; D.I. Srolovitz and G.N. Hassold, Phys. Rev. B 35 (1987) 6902. [I21 J. Villain, 1. Physique 46 (1985) 1843; D. Andelman and J.-F, Joanny, Phys. Rev. B 32 (1985) 4818; D.S. Fisher, Phys. Rev. Lett. 56 (1986) 419; D.A. Huse. Phys. Rev. B 36 (1987) 5383. [ 131 B.I. Halperin, PC. Hohenberg and S.-K. Ma, Phys. Rev. B 13 (1976) 4119. [ 141 T. Kawakatsu and K. Kawasaki, J. Colloid Interface Sci. 145 (1991) 420. [ 151 PC Martin, E.D. Siggia and H.H. Rose, Phys. Rev. A 8 (1973) 423; H.K. Janssen, Z Physik B 23 (1976) 377; R. Bausch, H.K. Janssen and H. Wagner, Z. Physik B 24 (1976) 113.
1978)