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Influence of temporal surface effects on the asymptotic behaviour of the nucleation-and-growth phenomena in some biopolymeric systems
Vacuum/volume
Pergamon PII : SOO42-207X(98)00021-9
50lnumber
I-2Ipages 79 to 83fl998 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0042-207X/98 $19.00+.00
0
influence of temporal surface effects on the asymptotic behaviour of the nucleation-and-growth phenomena in some biopolymeric systems Adam Gadomski,* University of Technology and Agriculture, Institute of Mathematics and Physics, Department of Theoretical Physics, Prof. Kaliskiego Street 7, PL-85796 Bydgoszcz, Poland
Surface or interface effects with temporal memories may even strikingly change the kinetic behaviour of the growing system. It turns out that the modified normal grain growth approach working in a regime of the long (fat)-tail kinetics offers a proper detection of the interfacial temporal changes on the system’s behaviour while a modified Mu//ins-Sekerka instability concept can rather hardly feel presence of a memory effect at the interface. Some extension of the approach to a nonequilibrium regime and/or application of a noise can change this situation to some extent. 0 7998 Elsevier Science Ltd. All rights reserved
1. Introduction Complexity of temporal phenomena taking place at solid-liquid interfaces and/or crystal surfaces have attracted permanent attention during the last decades. A considerable amount of information has accumulated about both the experimental methods as well as theoretical treatments developed for systems, where the surface represents the interface between a possibly ordered and non-liquid object and its non-solid as well as rather disordered environment.‘.’ In recent years, a substantial progress has been observed in applying high-quality experimental techniques (e.g., small- and wide-angle X-ray or neutron scattering ; laser Michelson interferometry ; quasi-elastic light scattering), mostly designed for performing dynamic or kinetic studies on the evolution or growth of some “soft-matter” systems, like protein crystals, model lipid membranes (i.e., biophysical systems), polymers or colloids. Particular examples may include studies on growth of lyzosyme (model protein) crystal? or crystallization of Aspergillus niger acid proteinase A,4 growth of quasi-crystalline domains in lipid bilayers (like DPPC or DOPE),’ formation of spherulites or cylindrolites (PE, nylon 6)h and slow colloidal aggregation (e.g., crystallization of PMMA particles).’ Theoretical considerations, in turn, stress pivotal role of nonequilibrium boundary conditions, importance of external field(s) controlling the growing process, influence of fluctuations at the interface on the system’s evolution ; also, smoothness (order)
* Presented at the 6th International Workshop on Electronic Properties of Metal/Non-Metal Microsystems, 8-12 September 1997, Prague, Czech Republic.
against roughness (disorder) phase transitions, interactions with solvent, presence of the depletion zone near a crystal face, surface diffusion and adsorption effects, etc., cannot be neglected here.‘.“‘O In this work, we wish to perform a comparative study towards kinetics of the growing processes established on one as well as on many (a large “statistical” number) crystallization seeds. In the former, where usually a part of the system is transformed into a new (crystal) phase, we will take into action a modified Mullins-Sekerka (M-S)-formalism, mostly due to Caroli and coworkers”,” and also considered in some recent work&* which assumes that the process of the single crystal formation from a supersaturated solution proceeds not only in an extremely slow way (because of the Laplacian concentration field being effectively used6.“.‘2), but also when the system in question is highly viscous and fluctuating which is a fundamental characteristic of many (bio)polymeric solutions except, perhaps, of those very diluted.3,‘0 In the latter, where the whole parent phase is transformed into a children crystalline phase, it is presumed that the (bio)polymeric system under study is a material composed of a possibly large number of crystallites that exchange the surface molecules among them. To be more precise, such an assumption is borrowed from the theory of the normal grain growth (NGG) which was applied successfully in the case of metal alloys and ceramics” as well as model biomaterials (a spatio-temporal version of the well-known Kolmogorov-Avrami kinetic equation has been proposed).14 In the NGG-concept which is a kind of nucleation-and-growth mechanism, it is assumed that a certain system is partitioned into small subunits called grains (crystallites). The main physical assumption here is that the system in question is cooperative 79
A Gadomski:
Nucleation-and-growth
phenomena
in some biopolymeric
since grains constantly exchange surface atoms, ions or molecules among them. Some additional condition has to be associated with that kind of growth, namely, that no single grain is allowed to dominate the whole resulting structure and no new grain may nucleate when a certain disappears. Let us underline here. according to what has been reported5.‘3.‘4 that the driving force of the process is a decrease of the total grain boundary energy (it is achieved by the system of constant volume, by shrinking of small crystallites and “expanding” some big ones). We wish to stress that in such a system fluctuations of the boundaries play a crucial role.‘4 There are, however, two basic physical presumptions which make, in this case, our approach markedly different from the NGG-description. These are formulated in Ref. [14] and are pertinent to kinetics of the crystallization within the material as well as adsorption at the interfaces crystal-external medium that undergo either the Avrami kinetics or manifest a non-Debyean relaxation behaviour, respectively (cf Ref. [ 141 and Refs therein). The two approaches proposed have much in common, but mostly. that they “produce” a fine polycrystalline or at least somehow ordered structure.“’ As being preferentially applied to some “soft-matter” systems (viz. protein crystals or amphiphilic assemblies of nanometer size) they show some differences due to presence of the depletion zone around a crystal,‘.‘“.“’ number and type of the crystallization centres considered, distribution of the viscoelastic properties over the whole system,” existence of other interfaces”.‘4.‘6 or influence of fluctuations of thermal as well as athermal nature on the system behaviou?” (a much recently explored concept of the fluctuation-driven transport termed as the ratchet device can be particularly useful here”). The paper is organized as follows. In the next section, a short presentation of the aforementioned approaches is given, and main results are provided. The last section offers final remarks. In this study, however, because of sake of brevity, we wish to present the key points of the modelling performed, paying rather minor attention to certainly very important details that one may find elsewhere.’ ” 2. Fundamentals of the modelling and results In the preceding section, we have mentioned that we will be trying to perform a kind of comparative study, based on modified MS- as well as NGG-approaches. The modification of the M-Sdescription in a highly viscous fluid “.” is due to the presumption that the interfacial tension, prescribed at the interface between a growing spherical object and external medium, is time-dependent. This effect was noticed many years ago for some polymeric solutions, where n-amyl or n-octyl alcohols played the role of solutes, and was evidenced even last year for some globular (e.g. ribonuclease) or large (serum albumin) protein adsorption. It was explained by means of a diffusion-controlled adsorption mechanism in which a change in the interfacial tension (or a change in the Gibbs capillary factor), proportional to the amount of macromolecules adsorbed, is time-dependent.” Also, a prescription of the nonequilibrium boundary condition at the crystal surface as well as some application even of a white Gaussian noise incorporated in the equation of interface motion may lead to some modification of the kinetic rules for a possibly (bio)poof the NGGlymeric crystalline system.6,x.‘h The modification approach, in turn, relies on presumption that the long (fat)-tail kinetics” of the process is to be used effectively which results in the diffusion-migration coefficient for the grain distribution in a material (of constant volume V) being time-dependent.s,‘4 80
systems
2.1. MullinsSekerka (M-S) approach with time dependent nonequilibrium interfacial condition. Since the M-S-formalism is really well-known, and is described elsewhere,6,“.‘2 let us briefly sketch the key points. An ideal sphere (for simplicity) is immersed in a Laplacian (stationary) concentration field which is prescribed at the internal (active interface) as well as external (“infinite”) boundaries. After the mass conservation law is being fulfilled, starting from a certain value of critical nucleus R*, the spherical object grows from the supersaturated solution by aggregation of Brownian particles, and an interplay between the external (diffusion) feeding field as well as the internal (capillary forces) field is observed during the whole process.“.6.2” We are interested in the evolution equation for the object” which spreads out with the following velocity
R’ = gj’(R,
t),
where D stands for the constant diffusion coefficient, s is the supersaturation (a parameter between 0 and l), R’ represents the ordinary first order derivative of R = R(t) with respect to time t, andf is of the form
f(R 1)=
1 - (R*/R) 1 +(R,/R)2
’
where R,, stands for a hydrodynamic radius. The only novel but physically expected’8,‘“,‘m5 thing that we wish to postulate here [i.e. concerning eqn (I)] is that R* is, according to the diffusioncontrolled adsorption mechanism assumed [crystal boundary as a perfect sink or, equivalently, a minimal energy barrier to adsorption, but, on the other hand no arrangements and/or conformational changes of the adsorbed macromolecules are taken into account by eqn (3) stated below]‘* to be time-dependent
R” = Ro-c(Dt)“‘,
(3)
where R, is the “equilibrium” nucleation radius,” and c is, for a given isoelectric point, a strongly temperatureas well as pHdependent parameter ;‘2X’8note, formally, that the determination of eqn (l), based on the M-S-formalism, allows such an incorporation of a time-dependent function, like that stated in eqn (3).6.2” Thus, in consequence of assumption (3), if one does not wish to violate the basic growth or aggregation condition here, i.e., that always R’ > 0, one has to conclude [see the form of eqns (I )-(2)] that, for the long time limit
R(t) cc t”’ ‘.?,
(4)
with v being rather small (v CCl/2) and depending upon the hydrodynamics” as well as thermodynamics” of the process in question. A determination of v, however, is not possible in a strict way, because some separation of R and t-variables in eqn (1) makes formal difliculties. Thus, we finish this subsection with the conclusion that some presence of a time-dependent condition at the interface changes somehow the scaling law (4) characterizing the process, but is, in fact, an only minor correction to the diffusional power law which holds for v = 0. Notice, also, that in the small viscosity limit one gets the classical M-S-system. A reasonable modification here can be due to a prescription of the nonequilibrium boundary condition and/or to some extension of the M-S-description towards incorporating a noise into the equation of the interface motion. The former, equivalent to
A Gadomski:
Nucleation-and-growth
phenomena
in some biopolymeric
f(R, t) cc R (see Ref. [6] for details), as well as the latter (if a splitting like R’ = If,,+ V,,, where Vr, as well as V,,, are deterministic and stochastic parts of the interface velocity, respectively, is invented8) may distinctly change the value of v producing a maximum value for v = l/2 so that a linear asymptotic behaviour R vs t can be approached.
systems
instants, is presented in Figs 1 and 2. For the first case for which the system evolves with the diffusion coefficient being directly proportional to the overall crystallization rate and inversely proportional to the adsorption rate, one gets, utilizing eqns (7H lo), the following asymptotic rule, namely R(t) cc t71s,
2.2. Normal Grain Growth (NGG) approach in a long (fat)-tail kinetic regime. The approach is based on the random walk model F,(n, t) = [N,
MO,
0L
(5)
(there exists obviously an equivalent stochastic differential equation for eqn (5) with the diffusion function (6) defined below; note that both equations are asymptotically stable”), where all the subscripts denote partial differentiations with respect to t and I,, respectively, F(v, t) stands for the distribution of grains of volume v in a total volume V. D(v, t) is the diffusion function which is deduced to be5,14of the power form of time
which comes from some scaling argumentation as well as from a certain linearization performed on two quantities: the global crystalization rate of the system and its adsorption rate (both are exponential in time, but none of them is a simple exponential ; the former is a “squeezed” exponential with B = 3, for a low temperature case,14 whereas the latter represents a stretched exponential with b lying between 0 and l/2 ; the case with b = l/2 characteristic of a most regular crystalline or grain shape will be chosen in our further considerations22) ; note that v213 corresponds to the surface of an individual grain. Other details of the process are described in Refs [14, 51. Applying a volume conservation law, typical for such systems, namely
(11)
whereas in the second case (B = 0) the only dominating “force” is the adsorption rate. 22Thus , following the same reasoning, one may get a drastic departure from more or less expected diffusional “one-half’ asymptotic rule, R(t) cc t”*,
(12)
which means a very slow growth. Surprisingly, note that an intermediate case which “interpolates” between the two afore presented, possesses diffusional kinetics, similar to that described by eqn (4). It is interesting that after applying a quite non-trivial formalism and using rather sophisticated as well as adequate presumptions as to time dependence of the diffusion (random
F(v,
tf
1.
0. 0.
0. 0.
2
(uO(t)) cc (tJ(t))R-3,
4
6
8
10
(7)
where (v’(t)), i = 0, 1, are zeroth and first moments of the process,s,‘4 respectively, and the size of the average radius of the crystallite is designated by R, again. The first moment is just the total (constant) volume of the system, V, but the zeroth moment is given by R(t)
(z+)(r)) cc u-3/4(r),
(8)
where
1.751 1.5
U(t)=-1 &b+l
1.25
P-f’.
Rule of type (4) is possible to get when one knows the solution to the problem (5) of the form5,‘4
-
9v4’3
F(u, t) = av”3u-7’4(t)exp
~
[ 16&(t)
1 ’
(10)
where a and C are positive constants,5 and u(t) is stated in eqn (9). Some visualization of solution F(u, t), with respect to an “order” (B = 3 ; a tendency of the system to get a minimum entropy state prevails) as well as a “disorder” (B = 0; the macromolecular adsorption takes favour) cases and for a few time
Figure 1. Dependence of the distribution function F(v, I) upon volume a of the individual crystallite [eqn (lo)] for three time moments t = 1.25, 1.5 and 1.75 from top to bottom, respectively, in the ordered case (B = 3 ; a = C = 1), where there is “a dynamic equilibrium” between the rate of propagation of the growing front in an ordered medium and the rate of adsorption of macromolecules at surfaces (at the top); the scaling law R(t) against t [eqn (1 l)] is presented at the bottom of the figure. 81
A Gadomski:
Nucleation-and-growth
phenomena
in some biopolymeric
I
V 2
6
4
8
10
R(t)
::;1-_0.41
systems
case a fully time-dependent stochastic approach, based on the NGG-description and believed to hold for highly viscous systems has concisely been utilized. The effect of such a work is as follows. Namely, applying the former, but with no nonequilibrium kinetic boundary condition prescribed at the interface23 and/or no “stochastic field” present,6.8 one is exclusively able to get a small departure from the diffusional behaviour of the asymptotic form given roughly by eqn (4) (strictly speaking, for v = 0). Utilizing the latter, in turn, one realizes quite big departures from that standard behaviour [see eqns (11) and (12)], but “on average”, one may expect again the well-known classical case. The conclusion would be that it is probably insufficient to presume the explicit time-dependence on the interface (that of a sphere from M-S-approach) or on the active surface (that belonging to a picked up crystallite treated by NGG-formalism), but one has to “put into play” a formalism which is as effective as possible, that means, that M-S-approach has to be properly modified (see Refs [6, 81). For similar reasons, the second mechanism (based on the NGG-concept) is probably more suitable for describing such quite complex and memory-feeling phenomena presented here, generally belonging to the class of polycrystal-monocrystal heterogeneous phase transitions. A common rationale, however, for performing such a comparative study is subject to very frequent use of the two above mentioned approaches to describe a temporal (or a spatio-temporal) behaviour of some “soft-matter” systems, like protein crystals or quasi-crystalline lipid domains, or even spherulites (cf Refs [4, 5, 10, 14, 18]), with a predominant role of the interface(s) present in the system.
0.2
I
t
c 0.5
1
1.5
2
Figure 2. Same as in Fig. 1, but for B = 0 (a = C = I), i.e. when the only
dominating mechanism is the adsorption rate (at the top). Equation (12) is shown at the bottom of the picture.
Acknowledgment The author wishes to thank the Department of Theoretical Physics and the Institute of Mathematics and Physics of the Technical University of Technology and Agriculture in Bydgoszcz (especially Dr Wojciech Chmara) for financial support. References
migration) function D(v, t),‘4.5 we have arrived here (note that the mean arithmetic value of l/S and 7/8 is just l/2), at the very classical (diffusional) behaviour, like in the simple and “protetic” M-S-approach. Note, however, that both the approaches presented may have much in common if they are modified properly or extended into stochastic domain.5~b~8~‘4 They both are able to show some interplay between an interface (adsorption, chemical reactionband a diffusion (also, mass convection) controlled growth behaviour characteristic of (bio)polymeric crystalline systems. They can also be useful for considering, in a limiting case, the kinetic behaviour of a thin film deposited on the solid support, mostly, when the filmI as well as the support*’ are taken into account in a certain separate way, i.e. no (or an extremely weak) interaction is assumed to hold between them. 3. Final remarks
In this work, we have presented a study (provided in a way out of traditional surface physics studies, rather) of the asymptotic evolutional behaviour of a viscous as well as possibly fluctuating (bio)polymeric system which has been treated twofold. In the first treatment, as a basis, a modified M-S-approach with an explicitly time-dependent [eqn (3)] or nonequilibrium internal boundary condition has been introduced whereas in the second 82
1. Woodruf, D. P. and Delchar, T. A., Modern Techniques af’Surfhce Science. Cambridge University Press, 1986. 2. Barab&si, A. L. and Stanley, H. E., Fractal Concepts in Surface Growth. Cambridge University Press, 1995. 3. Pusey, M. L., Snyder, R. S. and Naumann, R., J. Bial. Chem., 1986, 261,6524; Kam, Z., Shore, H. B. and Feher, G., J. Mol. Bial., 1978, 123, 539. 4. Kudo, N., Ataka, M., Sasaki, H., Muramatsu, T., Katsura, T. and Tanokura, M., J. Crystal Growth, 1996, 168, 118. 5. Gadomski, A., Phil. Mag. Left., 1994, 70, 335; Gadomski, A., tuczka, J., Kriechbaum, M., Laggner, P. and Jamnik, A., Phys. Lett. A, 1995,203,367. A. and tuczka, J., Inter. J. Quantum Chem., 1994, 52, 6. Gadomski, 301; Gadomski, A. and Luczka, J., in Far-From-Equilibrium Dynamics of‘ Chemical Systems, ed. J. Gbrecki, A. S. Cukrowski, A. S. Kawczynski and B. Nowakowski. World Scientific, 1994, pp. 342349. 7. Heymann, A., Stipp, A. and Schltzel, K., II Nuava Cimenta D, 1994, 16, 1149. 8. Gadomski, A. and tuczka, J., Acta Phys. Pal. B, 1993, 24, 725. 9. Somfai, E., Wolf, D. E. and Kertesz, J., J. Phys. I. France, 1996, 6, 393; Shimon, L. J. W., Vaida, M., Addadi, L., Lahav, M. and Leiserovitz, L., J. Am. Chem. Sac., 1990, 112,62 15. 10. Atakd, M., Prag. Crystal Growth and Charact., 1995, 30, 109; Vekilow, P. G., Ataka, M. and Katsura, T., Acfa Cryst. D, 1995,51,207. Il. Langer, J. S., Reti. Mad. Phys., 1980, 52, 1 ; Mullins, W. W. and Sekerka, R. F., J. Appl. Phys., 1963,34,323. 12. Caroli, B., Caroli, C., Roulet, B. and Faivre, G., J. Crystal Growth, 1989, 94,253.
A Gadomski:
Nucleation-and-growth
phenomena
in some biopolymeric
13. Chen, L. Q. and Yang, W., Phys. Rev. B, 1994,50,15752. A.. Chem. Phvs. Lett.. 1996. 258. 6: Gadomski. A., 14. Gadomski. Berichte Binsenges Phys. them., 1946, lo& 134: 15. Schmelzer, J. and Gutzow, I., Restocker Physik Munuskr., 1989, 13, 82. 16. Vandewalle, N., Ausloos, M. and Cloots, R., J. Crystal Growth, 1996, 169,79. 17. Ajdari, A., Europhys. Letr., 1995, 31,69; Harms, T. and Lipowsky, R., Phys. Rev. Lett., 1997, 79, 2895. 18. Ward, A. F. H. and Tordai, L., 1. Chem. Phys., 1946, 14,453 ; Van der Vegt, W., Norde, W., Van der Mei, H. C. and Busscher, H. J., Colloid PO&m. Sri., 1996, 274, 27.
systems
19. Nagle, J. F., Biophys. J., 1992,63,366; Kopelman, R., Science, 1988, 241, 1620; Plonka, A., Kroh, J. and Berlin, Yu. A., Chem. Phys. Left., 1988, 153,433. 20. tuczka, J., Gadomski, A. and Grzywna, 2. J., IMA Preprint Series No. 869. University of Minnesota U.S.A., 1991 ; Gadomski, A., Grzywna, Z. J. and Luczka, J. Chem. Engng Sci., 1993,48,3713 ; Danch, A. and Gadomski, A., J. Thermal Analysis, 1995,45, 1175. 21. Rudnicki, R., J. IXfferentiul Eqs, 1993, 102, 391. 22. Johnson, H. E., Douglas, J. F. and Granick, S., Ph_vs. Rev. Left., 1993,70,3267. 23. Goldenfeld, N., J. Crystal Growth, 1987, 84, 601.
83
Pergamon PII : SOO42-207X(98)00021-9
50lnumber
I-2Ipages 79 to 83fl998 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0042-207X/98 $19.00+.00
0
influence of temporal surface effects on the asymptotic behaviour of the nucleation-and-growth phenomena in some biopolymeric systems Adam Gadomski,* University of Technology and Agriculture, Institute of Mathematics and Physics, Department of Theoretical Physics, Prof. Kaliskiego Street 7, PL-85796 Bydgoszcz, Poland
Surface or interface effects with temporal memories may even strikingly change the kinetic behaviour of the growing system. It turns out that the modified normal grain growth approach working in a regime of the long (fat)-tail kinetics offers a proper detection of the interfacial temporal changes on the system’s behaviour while a modified Mu//ins-Sekerka instability concept can rather hardly feel presence of a memory effect at the interface. Some extension of the approach to a nonequilibrium regime and/or application of a noise can change this situation to some extent. 0 7998 Elsevier Science Ltd. All rights reserved
1. Introduction Complexity of temporal phenomena taking place at solid-liquid interfaces and/or crystal surfaces have attracted permanent attention during the last decades. A considerable amount of information has accumulated about both the experimental methods as well as theoretical treatments developed for systems, where the surface represents the interface between a possibly ordered and non-liquid object and its non-solid as well as rather disordered environment.‘.’ In recent years, a substantial progress has been observed in applying high-quality experimental techniques (e.g., small- and wide-angle X-ray or neutron scattering ; laser Michelson interferometry ; quasi-elastic light scattering), mostly designed for performing dynamic or kinetic studies on the evolution or growth of some “soft-matter” systems, like protein crystals, model lipid membranes (i.e., biophysical systems), polymers or colloids. Particular examples may include studies on growth of lyzosyme (model protein) crystal? or crystallization of Aspergillus niger acid proteinase A,4 growth of quasi-crystalline domains in lipid bilayers (like DPPC or DOPE),’ formation of spherulites or cylindrolites (PE, nylon 6)h and slow colloidal aggregation (e.g., crystallization of PMMA particles).’ Theoretical considerations, in turn, stress pivotal role of nonequilibrium boundary conditions, importance of external field(s) controlling the growing process, influence of fluctuations at the interface on the system’s evolution ; also, smoothness (order)
* Presented at the 6th International Workshop on Electronic Properties of Metal/Non-Metal Microsystems, 8-12 September 1997, Prague, Czech Republic.
against roughness (disorder) phase transitions, interactions with solvent, presence of the depletion zone near a crystal face, surface diffusion and adsorption effects, etc., cannot be neglected here.‘.“‘O In this work, we wish to perform a comparative study towards kinetics of the growing processes established on one as well as on many (a large “statistical” number) crystallization seeds. In the former, where usually a part of the system is transformed into a new (crystal) phase, we will take into action a modified Mullins-Sekerka (M-S)-formalism, mostly due to Caroli and coworkers”,” and also considered in some recent work&* which assumes that the process of the single crystal formation from a supersaturated solution proceeds not only in an extremely slow way (because of the Laplacian concentration field being effectively used6.“.‘2), but also when the system in question is highly viscous and fluctuating which is a fundamental characteristic of many (bio)polymeric solutions except, perhaps, of those very diluted.3,‘0 In the latter, where the whole parent phase is transformed into a children crystalline phase, it is presumed that the (bio)polymeric system under study is a material composed of a possibly large number of crystallites that exchange the surface molecules among them. To be more precise, such an assumption is borrowed from the theory of the normal grain growth (NGG) which was applied successfully in the case of metal alloys and ceramics” as well as model biomaterials (a spatio-temporal version of the well-known Kolmogorov-Avrami kinetic equation has been proposed).14 In the NGG-concept which is a kind of nucleation-and-growth mechanism, it is assumed that a certain system is partitioned into small subunits called grains (crystallites). The main physical assumption here is that the system in question is cooperative 79
A Gadomski:
Nucleation-and-growth
phenomena
in some biopolymeric
since grains constantly exchange surface atoms, ions or molecules among them. Some additional condition has to be associated with that kind of growth, namely, that no single grain is allowed to dominate the whole resulting structure and no new grain may nucleate when a certain disappears. Let us underline here. according to what has been reported5.‘3.‘4 that the driving force of the process is a decrease of the total grain boundary energy (it is achieved by the system of constant volume, by shrinking of small crystallites and “expanding” some big ones). We wish to stress that in such a system fluctuations of the boundaries play a crucial role.‘4 There are, however, two basic physical presumptions which make, in this case, our approach markedly different from the NGG-description. These are formulated in Ref. [14] and are pertinent to kinetics of the crystallization within the material as well as adsorption at the interfaces crystal-external medium that undergo either the Avrami kinetics or manifest a non-Debyean relaxation behaviour, respectively (cf Ref. [ 141 and Refs therein). The two approaches proposed have much in common, but mostly. that they “produce” a fine polycrystalline or at least somehow ordered structure.“’ As being preferentially applied to some “soft-matter” systems (viz. protein crystals or amphiphilic assemblies of nanometer size) they show some differences due to presence of the depletion zone around a crystal,‘.‘“.“’ number and type of the crystallization centres considered, distribution of the viscoelastic properties over the whole system,” existence of other interfaces”.‘4.‘6 or influence of fluctuations of thermal as well as athermal nature on the system behaviou?” (a much recently explored concept of the fluctuation-driven transport termed as the ratchet device can be particularly useful here”). The paper is organized as follows. In the next section, a short presentation of the aforementioned approaches is given, and main results are provided. The last section offers final remarks. In this study, however, because of sake of brevity, we wish to present the key points of the modelling performed, paying rather minor attention to certainly very important details that one may find elsewhere.’ ” 2. Fundamentals of the modelling and results In the preceding section, we have mentioned that we will be trying to perform a kind of comparative study, based on modified MS- as well as NGG-approaches. The modification of the M-Sdescription in a highly viscous fluid “.” is due to the presumption that the interfacial tension, prescribed at the interface between a growing spherical object and external medium, is time-dependent. This effect was noticed many years ago for some polymeric solutions, where n-amyl or n-octyl alcohols played the role of solutes, and was evidenced even last year for some globular (e.g. ribonuclease) or large (serum albumin) protein adsorption. It was explained by means of a diffusion-controlled adsorption mechanism in which a change in the interfacial tension (or a change in the Gibbs capillary factor), proportional to the amount of macromolecules adsorbed, is time-dependent.” Also, a prescription of the nonequilibrium boundary condition at the crystal surface as well as some application even of a white Gaussian noise incorporated in the equation of interface motion may lead to some modification of the kinetic rules for a possibly (bio)poof the NGGlymeric crystalline system.6,x.‘h The modification approach, in turn, relies on presumption that the long (fat)-tail kinetics” of the process is to be used effectively which results in the diffusion-migration coefficient for the grain distribution in a material (of constant volume V) being time-dependent.s,‘4 80
systems
2.1. MullinsSekerka (M-S) approach with time dependent nonequilibrium interfacial condition. Since the M-S-formalism is really well-known, and is described elsewhere,6,“.‘2 let us briefly sketch the key points. An ideal sphere (for simplicity) is immersed in a Laplacian (stationary) concentration field which is prescribed at the internal (active interface) as well as external (“infinite”) boundaries. After the mass conservation law is being fulfilled, starting from a certain value of critical nucleus R*, the spherical object grows from the supersaturated solution by aggregation of Brownian particles, and an interplay between the external (diffusion) feeding field as well as the internal (capillary forces) field is observed during the whole process.“.6.2” We are interested in the evolution equation for the object” which spreads out with the following velocity
R’ = gj’(R,
t),
where D stands for the constant diffusion coefficient, s is the supersaturation (a parameter between 0 and l), R’ represents the ordinary first order derivative of R = R(t) with respect to time t, andf is of the form
f(R 1)=
1 - (R*/R) 1 +(R,/R)2
’
where R,, stands for a hydrodynamic radius. The only novel but physically expected’8,‘“,‘m5 thing that we wish to postulate here [i.e. concerning eqn (I)] is that R* is, according to the diffusioncontrolled adsorption mechanism assumed [crystal boundary as a perfect sink or, equivalently, a minimal energy barrier to adsorption, but, on the other hand no arrangements and/or conformational changes of the adsorbed macromolecules are taken into account by eqn (3) stated below]‘* to be time-dependent
R” = Ro-c(Dt)“‘,
(3)
where R, is the “equilibrium” nucleation radius,” and c is, for a given isoelectric point, a strongly temperatureas well as pHdependent parameter ;‘2X’8note, formally, that the determination of eqn (l), based on the M-S-formalism, allows such an incorporation of a time-dependent function, like that stated in eqn (3).6.2” Thus, in consequence of assumption (3), if one does not wish to violate the basic growth or aggregation condition here, i.e., that always R’ > 0, one has to conclude [see the form of eqns (I )-(2)] that, for the long time limit
R(t) cc t”’ ‘.?,
(4)
with v being rather small (v CCl/2) and depending upon the hydrodynamics” as well as thermodynamics” of the process in question. A determination of v, however, is not possible in a strict way, because some separation of R and t-variables in eqn (1) makes formal difliculties. Thus, we finish this subsection with the conclusion that some presence of a time-dependent condition at the interface changes somehow the scaling law (4) characterizing the process, but is, in fact, an only minor correction to the diffusional power law which holds for v = 0. Notice, also, that in the small viscosity limit one gets the classical M-S-system. A reasonable modification here can be due to a prescription of the nonequilibrium boundary condition and/or to some extension of the M-S-description towards incorporating a noise into the equation of the interface motion. The former, equivalent to
A Gadomski:
Nucleation-and-growth
phenomena
in some biopolymeric
f(R, t) cc R (see Ref. [6] for details), as well as the latter (if a splitting like R’ = If,,+ V,,, where Vr, as well as V,,, are deterministic and stochastic parts of the interface velocity, respectively, is invented8) may distinctly change the value of v producing a maximum value for v = l/2 so that a linear asymptotic behaviour R vs t can be approached.
systems
instants, is presented in Figs 1 and 2. For the first case for which the system evolves with the diffusion coefficient being directly proportional to the overall crystallization rate and inversely proportional to the adsorption rate, one gets, utilizing eqns (7H lo), the following asymptotic rule, namely R(t) cc t71s,
2.2. Normal Grain Growth (NGG) approach in a long (fat)-tail kinetic regime. The approach is based on the random walk model F,(n, t) = [N,
MO,
0L
(5)
(there exists obviously an equivalent stochastic differential equation for eqn (5) with the diffusion function (6) defined below; note that both equations are asymptotically stable”), where all the subscripts denote partial differentiations with respect to t and I,, respectively, F(v, t) stands for the distribution of grains of volume v in a total volume V. D(v, t) is the diffusion function which is deduced to be5,14of the power form of time
which comes from some scaling argumentation as well as from a certain linearization performed on two quantities: the global crystalization rate of the system and its adsorption rate (both are exponential in time, but none of them is a simple exponential ; the former is a “squeezed” exponential with B = 3, for a low temperature case,14 whereas the latter represents a stretched exponential with b lying between 0 and l/2 ; the case with b = l/2 characteristic of a most regular crystalline or grain shape will be chosen in our further considerations22) ; note that v213 corresponds to the surface of an individual grain. Other details of the process are described in Refs [14, 51. Applying a volume conservation law, typical for such systems, namely
(11)
whereas in the second case (B = 0) the only dominating “force” is the adsorption rate. 22Thus , following the same reasoning, one may get a drastic departure from more or less expected diffusional “one-half’ asymptotic rule, R(t) cc t”*,
(12)
which means a very slow growth. Surprisingly, note that an intermediate case which “interpolates” between the two afore presented, possesses diffusional kinetics, similar to that described by eqn (4). It is interesting that after applying a quite non-trivial formalism and using rather sophisticated as well as adequate presumptions as to time dependence of the diffusion (random
F(v,
tf
1.
0. 0.
0. 0.
2
(uO(t)) cc (tJ(t))R-3,
4
6
8
10
(7)
where (v’(t)), i = 0, 1, are zeroth and first moments of the process,s,‘4 respectively, and the size of the average radius of the crystallite is designated by R, again. The first moment is just the total (constant) volume of the system, V, but the zeroth moment is given by R(t)
(z+)(r)) cc u-3/4(r),
(8)
where
1.751 1.5
U(t)=-1 &b+l
1.25
P-f’.
Rule of type (4) is possible to get when one knows the solution to the problem (5) of the form5,‘4
-
9v4’3
F(u, t) = av”3u-7’4(t)exp
~
[ 16&(t)
1 ’
(10)
where a and C are positive constants,5 and u(t) is stated in eqn (9). Some visualization of solution F(u, t), with respect to an “order” (B = 3 ; a tendency of the system to get a minimum entropy state prevails) as well as a “disorder” (B = 0; the macromolecular adsorption takes favour) cases and for a few time
Figure 1. Dependence of the distribution function F(v, I) upon volume a of the individual crystallite [eqn (lo)] for three time moments t = 1.25, 1.5 and 1.75 from top to bottom, respectively, in the ordered case (B = 3 ; a = C = 1), where there is “a dynamic equilibrium” between the rate of propagation of the growing front in an ordered medium and the rate of adsorption of macromolecules at surfaces (at the top); the scaling law R(t) against t [eqn (1 l)] is presented at the bottom of the figure. 81
A Gadomski:
Nucleation-and-growth
phenomena
in some biopolymeric
I
V 2
6
4
8
10
R(t)
::;1-_0.41
systems
case a fully time-dependent stochastic approach, based on the NGG-description and believed to hold for highly viscous systems has concisely been utilized. The effect of such a work is as follows. Namely, applying the former, but with no nonequilibrium kinetic boundary condition prescribed at the interface23 and/or no “stochastic field” present,6.8 one is exclusively able to get a small departure from the diffusional behaviour of the asymptotic form given roughly by eqn (4) (strictly speaking, for v = 0). Utilizing the latter, in turn, one realizes quite big departures from that standard behaviour [see eqns (11) and (12)], but “on average”, one may expect again the well-known classical case. The conclusion would be that it is probably insufficient to presume the explicit time-dependence on the interface (that of a sphere from M-S-approach) or on the active surface (that belonging to a picked up crystallite treated by NGG-formalism), but one has to “put into play” a formalism which is as effective as possible, that means, that M-S-approach has to be properly modified (see Refs [6, 81). For similar reasons, the second mechanism (based on the NGG-concept) is probably more suitable for describing such quite complex and memory-feeling phenomena presented here, generally belonging to the class of polycrystal-monocrystal heterogeneous phase transitions. A common rationale, however, for performing such a comparative study is subject to very frequent use of the two above mentioned approaches to describe a temporal (or a spatio-temporal) behaviour of some “soft-matter” systems, like protein crystals or quasi-crystalline lipid domains, or even spherulites (cf Refs [4, 5, 10, 14, 18]), with a predominant role of the interface(s) present in the system.
0.2
I
t
c 0.5
1
1.5
2
Figure 2. Same as in Fig. 1, but for B = 0 (a = C = I), i.e. when the only
dominating mechanism is the adsorption rate (at the top). Equation (12) is shown at the bottom of the picture.
Acknowledgment The author wishes to thank the Department of Theoretical Physics and the Institute of Mathematics and Physics of the Technical University of Technology and Agriculture in Bydgoszcz (especially Dr Wojciech Chmara) for financial support. References
migration) function D(v, t),‘4.5 we have arrived here (note that the mean arithmetic value of l/S and 7/8 is just l/2), at the very classical (diffusional) behaviour, like in the simple and “protetic” M-S-approach. Note, however, that both the approaches presented may have much in common if they are modified properly or extended into stochastic domain.5~b~8~‘4 They both are able to show some interplay between an interface (adsorption, chemical reactionband a diffusion (also, mass convection) controlled growth behaviour characteristic of (bio)polymeric crystalline systems. They can also be useful for considering, in a limiting case, the kinetic behaviour of a thin film deposited on the solid support, mostly, when the filmI as well as the support*’ are taken into account in a certain separate way, i.e. no (or an extremely weak) interaction is assumed to hold between them. 3. Final remarks
In this work, we have presented a study (provided in a way out of traditional surface physics studies, rather) of the asymptotic evolutional behaviour of a viscous as well as possibly fluctuating (bio)polymeric system which has been treated twofold. In the first treatment, as a basis, a modified M-S-approach with an explicitly time-dependent [eqn (3)] or nonequilibrium internal boundary condition has been introduced whereas in the second 82
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Nucleation-and-growth
phenomena
in some biopolymeric
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systems
19. Nagle, J. F., Biophys. J., 1992,63,366; Kopelman, R., Science, 1988, 241, 1620; Plonka, A., Kroh, J. and Berlin, Yu. A., Chem. Phys. Left., 1988, 153,433. 20. tuczka, J., Gadomski, A. and Grzywna, 2. J., IMA Preprint Series No. 869. University of Minnesota U.S.A., 1991 ; Gadomski, A., Grzywna, Z. J. and Luczka, J. Chem. Engng Sci., 1993,48,3713 ; Danch, A. and Gadomski, A., J. Thermal Analysis, 1995,45, 1175. 21. Rudnicki, R., J. IXfferentiul Eqs, 1993, 102, 391. 22. Johnson, H. E., Douglas, J. F. and Granick, S., Ph_vs. Rev. Left., 1993,70,3267. 23. Goldenfeld, N., J. Crystal Growth, 1987, 84, 601.
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