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The kinetics of formation and morphological memory of polymeric nanostructures
I. V. Melikhov1, O. V. Alekseeva2, V.N. Rudin1, E.D. Kozlovskaya1, A. V. Noskov2 1M. V. Lomonosov Moscow State University, Moscow, Russia 2G. A. Krestov Institute of Solution Chemistry of RAS, Ivanovo, Russia melikhov@radio.chem.msu.ru, ova@isc-ras.ru
PACS 81.05.Lg DOI 10.17586/2220-8054-2015-6-4-551-561
The mesokinetic model of the polymeric body formation was formulated, describing the nucleation, growth and aggregation of macromolecules followed by the aggregates' assembly into the nanostructured body. According to this model, the near-surface nanostructure of the polymeric body retains the information on how it was formed, and this information can be extracted from the morphological characteristics of the microrelief of the body while maintaining its integrity, that is, body has the morphological memory. The mesokinetic model was used to study film formation by the evaporation of an o-xylene or toluene solution of polystyrene, which provided an opportunity to identify the kinetics of nanostructure formation found in the surface area of the film after completion of the process. It turned out that at the beginning of evaporation, a number of the primary macromolecules in the form of spheroidal nanoparticles were formed, and then, the primary aggregates coalesced into the secondary aggregates forming the ordered chains of nanoparticles. Keywords: polystyrene, microrelief, primary (nanoparticles) and secondary (microparticle) units. Received: 9 April 2015
1. Introduction
According to the electron micrographs of polymers (films, granules, membranes, etc.) published in the literature, many of these substances have nanostructured microrelief. We observed such microrelief on the surface of polystyrene granules purchased from "Aldrich" (Germany), as indicated by Fig. 1, which shows that the near-surface areas of the studied granules have a multilevel hierarchical nanostructure. They consist of aggregates of polystyrene molecules in the form of spheroidal nanoparticles, aggregated into the chains, combined into microparticles, which form a granule. Fig. 1 leads to the question, whether the microrelief of the polymeric body retains information about how it was formed. Is it possible, focusing on electron microscopic images of the microrelief, to determine the speed of the elementary processes that lead to this microrelief? As it is known, the exact answer to this question may be provided by studying the kinetics of elementary processes for monomer integration into oligomer and polymer molecules, the formation of aggregates from polymer molecules (associates, floccules, colloidal particles) and aggregates association in a polymeric body (solidification, crystallization) [1-8]. Methods of studying of these elementary processes have been developed and are widely used [3-11]. In this case, the design of experiments and interpretation of the results is based on kinetic models that enable the successful description of many experimental results [11-15], but do not allow their quantitative prediction, as each model inevitably contains arbitrary assumptions that reduce its predictive ability. With this in mind, this article attempts to formulate a mesokinetic model in which the only arbitrary assumption is the crucial role of binary collisions of particles in the system, and other assumptions are included in the unknown frequency functions and are available for
experimental determination. The proposed model was used to study the microrelief of the films formed by evaporation of a polystyrene solution (Fig. 2), which led to the concept of morphological film memory. The study of the microrelief of granules from Aldrich (Fig. 1) as well as the study of the electron-microscopic images of many polymeric bodies described in the literature has led to the same conclusion. This enabled us to assert that many polymeric bodies possess morphological memory which is different from the shape memory and other types of polymer memory [16-18] with the memory carrier being localized at the body surface, rather than within its volume.
1 Uiu
Fig. 1. Electron micrograph of the surface area of a typical polystyrene granules from "Aldrich" (Germany). Microscope CamScan, a sample was prepared by the standard method
2. Model of nanostructure formation
The processes in a homogeneous solution of a monomer A which evaporates at a predetermined speed are studied. At the time t = 0 the polymerization initiator B is added into the solution, which leads to the formation of polymeric molecules which combine together to form ordered aggregates. Upon evaporation of the solution, its volume decreases, leading to an increase in the concentration of molecules and aggregates, which do not evaporate. During the evaporation, an increasing number of the molecules and aggregates is brought into contact with each other, and at the moment tF of the complete removal of the solvent they form a polymer body with the following hierarchical structure, its nanolevel consisting of polymer molecules, microlevel - by the aggregates, and macrolevel - by the body itself.
At 0 < t < tF the solution contains monomer molecules (j = 0), polymer molecules (j = 1) and aggregates (j = 2, 3...).
The state of each particle is characterized by parameters X = {nl.. .ni... nm,v, Xl... Xi... Xp}, where ni is the number of A molecules, which have formed one of the m of polymer molecules of the particle, with a total number of the A molecules equal to n = ml ni; v is a particle volume; Xi is a parameter of shape, composition or structure of the particle. The condition for the conservation of the number of atoms introduced into the
Fig. 2. Microrelief of the film obtained by removing the solvent from the o-xylene solution of polystyrene. The conditions of film formation: V = 36, 5 cm3, T = 293K, initial concentration of polystyrene in the solution Mo = 0.15 g/cm3, S = 1200 cm2. Microscope CamScan
system in the molecules A is fulfilled:
caovo = /
Jy
Ca (1 - e) + ^^nNjn (t)
dV, (1)
where CA0 and CA are the initial and the current concentration of the A molecules in the solution; V0 and V are the initial and the current volume of the system; e is the share of the system volume occupied by the polymer molecules and aggregates; Njn (t) is the concentration of macromolecules and aggregates, including n molecules of A by the time t.
The volume V is equal to:
V = Vo / vlSusdt, (2)
0
where vL is the volume change after the evaporation of one of the solvent molecules; S is the area of the evaporation surface; uS is the flow of solvent molecules in the vapor at the moment t.
At 0 < t < tF the concentration Njn (t) in each system area changes due to the associations of the particles, because of the binary collisions and fragmentation of molecules and aggregates under the influence of thermal motion and mechanical effects. Therefore:
ANjn(t) = (j - j - divJj^j At, (3)
where ANjn(t) is the change of the Njn(t) between the time moments of t and t + At; Qj+ and Qj- are the frequency functions that characterize the processes that lead to an increase and decrease of n of the particles of jth type; Jjn is the flow of the particles in the condition (j, n) through the given area.
Therein:
Ca +
+ Y1 f3*
,n+bNk,n+bj (4)
k>j
Qj- = ajnNjnCA + ^ ajnNjnNk,a + 3jnNjn, (5)
k>j
where Njn = Njn(t); ajn and 3jn are functions that characterize the frequency of association and fragmentation of particles; a and b are the characteristic changes of n during a single act of unification and fragmentation.
With the known frequency functions ajn(n,t) and 3jn(n,t), equations (1)-(5) allow one to calculate the volume fraction of the area filled with the particles of jth type:
£j(t) = Yl uAjnNjn(t)j (6)
n
where VAj is the volume per single A molecule after its transition into the polymer molecule of the jth type particle.
From relations (4)-(6), it follows that moving from the concentration Njn(t) to the distribution function 'jn = dNjn/dn and expanding ajn'jn and 3jn'jn in Taylor series limiting to three terms of the expansion, equation (3) can be written as:
d^jn _ d2 (Djn 'jn) d (Gjn'jn)
dt dn2 dn + Wjk - dWj (7)
where Djn = 1 (a2ajnCA + b23jn); Gjn = aajnCA — b3jn. Wjk is the function of the transition of the particles of jth type in the groups of particles of other kth types, related by the frequency functions in accordance with the conditions (4) and (5). Therein:
£j (t) = VAj n'jn(t)dn, (8)
J n>njo
where nj0 is a number of molecules A, forming a particle of minimum size, to which the speed of enlargement Gjn can be attributed.
The solution of equation (7) with the corresponding boundary conditions can provide the ability to search the particle distribution function for a status parameter X, which
can be expressed as 'j = 'jn(t)fj (u)n fj (X)n .-¡j (X)nv ...fj (XP)nu, where fj (u)n
and fj (Xi)nv are the distribution density on v of the particles with the given n and the distribution density on Xi of the particles with the given n and u.
3. Morphological body memory model
Within the framework of the considered model, the microrelief of the body at time t is represented by a set of ordered aggregates, among which the molecules are located, with the degree of filling of the volume of the body's surface area by particles of:
£s = VAjn'jn (tF)S dn, (9)
J n>njo
where 'jn (tF)S is the distribution on n of particles of the jth type on the body's surface area at the moment tF. Distribution 'jn (tF)S, as well as any distribution on any area of the system, is an integral of equation (7) and is determined by which of the frequency functions ajn (n,tF) and 3jn (n,tF) occurred at t ^ tF. Consequently, the information about the functions ajn (n,tF) and 3jn (n,tF), included in this integral, can be extracted from the
distribution pjn (tF )S. Taking this into account, functions Pjn (tF )S can be considered as carriers of the body morphological memory of the process which during the time tF has led its microrelief into the state represented by the equation (9). According to equations (1)-(8), the frequency functions depend on the system properties yy = [ca0,v0, S,uS}, accordingly the microrelief saves the information about the conditions in which the body was formed.
The distribution Pjn (tF)S can be variable. For the formation of macromolecules not prone to aggregation in a homogeneous system under the condition of independence of ajn and /3jn, this distribution on n is determined by integrating the equation (7) in the form:
dPjn _ n d2P1n n dP1n ,1iV.
= D1n n 2--G1n^-. (10)
dt dn2 on
If the conditions y are such that a1nCA >> p1n, integration leads [19,20] to the formula
Mp)-1/2 exp (-X-) - exp (erfc (X+)
P1n (tF )S = CB0 where CB0 is the initiator concentration in the initial solution:
(11)
pi C\ rtt
A = Grndt; p = Dm/Gm; X± = (n - noj ± A) / (4nAp)1/2 ; erfc (X+) = / e-x2dX. Jo Vn Jx+
Equation (11) describes the molecular weight distribution of the polymer molecules forming the microrelief of the body, provided that the body was formed according to the model (1)-(8) in the independence of D1n and G1n on n. Given this, we can assume that in systems where the experimental data on the microrelief is described by relation (11), this condition is realized, and the information about it is accumulated by the distribution P1n (tF)S.
When t > tF, the distribution p1n (tF )S can change under the influence of various factors, so that going from n to X:
<Pj {X, r) = j (tF )S Pj (X, tF (12)
where Pj ^X, ^ is the distribution function on X of the near-surface particles of the jih
type after influences on the body over time r; Pj (n,tF, r) is the indicator of the microrelief stability to external influences.
Function Pj [xttF ,r^J, which characterizes the degree of preservation of the body's
morphological memory after its use, can be defined in the framework of our model after detailing of the respective frequency functions.
From the above, it follows that if after the formation and use of the body the microrelief remains on its surface, allowing one to experimentally determine the distribution functions of the structural elements of the microrelief on morphological parameters of the state and to present them as a solution of the equation (7), it can be argued that the body has morphological memory. According to the electron micrographs published in the literature, there are several polymeric bodies exhibiting morphological memory. Such a memory is also possessed by many inorganic nanostructured materials, although their nanoparticles, after aggregation, often become indistinguishable.
4. Experimental study of polystyrene film formation. Methods and results
In conducting this research, we used a previously-described technique [21,22]. The polystyrene granules from Aldrich were dissolved in a solvent capable of rapid evaporation under normal conditions. A portion of the solution with a predetermined polystyrene content was introduced into a flat glass cuvette, which was then placed in the thermostatic chamber of a scale for continuous weighing, so that the weight change of the cuvette with the solution during the evaporation of the solvent was observed. Evaporation was carried out so that only the molecules of the solvent evaporated, making it possible to determine the rate at which the concentration of the polystyrene solution increased by measuring the rate at which the mass of the cuvette decreased. When the mass of the cuvette stopped changing, the formed polymer film was removed from the cuvette, its near-surface areas were studied using a scanning electron microscope, and the distribution functions of particle images of its microrelief on size were determined: 0k (£) = Bk (£) /Bk0, where Bk (£) is the number of measured particles of kth type, that are smaller than £ = s1/2 at the area of the image of s; Bk0 is the total number of particles measured. The particles which had homogeneous images were taken for primary nanoparticles (k = 1), the ordered clusters of primary particles were taken for the aggregates (microparticles, k = 2), and the groups of orderly contacted assemblies were taken for the secondary aggregates (k = 3). In some experiments, a solution of Ceo was added to the initial solution of polystyrene, and then the action was taken as described above.
The experiments have shown that the rate of the solvent removal from the cuvette was varied in two steps, as can be seen in Fig. 3, which shows the data on weight changes of the toluene solution of polystyrene in the cuvette. Therein the condition was fulfilled:
J = JoA + Ji exp[—u (t — ti)](1 — A), (13)
where J0 and J1 are flows of evaporated solvent molecules from the surface of the cuvette at t < t1 and t = t1; A = 1 at t < t1 and A = 0 at t > t1; J0 = 2, 06 ■ 10-3mol/(m2 ■ s); J1 = 2, 53 ■ 10-3mol/(m2 ■ s); u = (2, 3 ± 0, 5) ■ 10-3c-1 is the empirical parameter.
By the end of the second step, when the evaporation was almost complete, a polymer film was formed in the cuvette which had properties that did not alter after transferring it to an electron microscope. Microrelief of the film was created by the microparticles, single or combined into the chains spheroidal nanoparticles were found on their surface and between them (Fig. 2). For the microparticles and their constituent nanoparticles the functions 6k (l) are presented as functions of Z, shown in Fig. 4, where Z = (l — lk0)/(lkM — lk0), lk0 is the minimum size of measured particles of kth type; lkM is the size corresponding to the condition @k(lkM) = 0, 5 (Table 1). The corresponding differential distribution functions Ak (l) = p are described by the formula:
1 (l-lk
Ak (l)
(nAkpk) 1/2 exp i—X2_) — -^exp ( -—— ) erfc (X+)
2pk V Pk )
(14)
where Ak, pk lk0 are empirical parameters, some of which are listed in the Table 2; X± = (l — lk0 ± Ak) / (4nAkPk)1/2.
Functions 91 (l) of nanoparticles, included in the microparticles of different sizes, were similar. They changed when fullerene was introduced into the system (Fig. 4). In particular, during the formation of the film using an o-xylene solution of polystyrene at V0 = 36, 5 cm3, M0 = 0,15 g/cm3 and initial fullerene concentration of MC = 10-5 —10-4 g/cm3, the function 91 (l) shifted toward larger units provided the following:
l1M = -m0 (1 + 5,1 ■ 102Mc/M0) , (15)
Fig. 3. Change of mass M of the polystyrene solution in toluene. The experimental conditions: solvent: toluene, T = 293 K, V0 = 0.463 cm3, initial solution weight M0 = 0.439 g, S = 15.2 cm2
Table 1. The characteristic size of the film nanostructure elements
Nanoparticle type Mc/Mo lko, nm lkM, nm
nano 0 21±2 45±1
micro 0 280±20 750±20
nano 3■10-4 24±2 53±2
nano 1■10-3 26±2 65±2
Table 2. The parameters of the differential distribution functions of nanopar-ticles microrelief on size
Body type lko,nm Ak ,nm Pk ,nm nio
film nanoparticle 21±2 43±1 2,0±0,2 (3,1±0,1)-108
film microparticle 280±40 690±10 30±3 (7,4±0,5)-109
where lMo is the size in the absence of fullerene. 5. Discussion
The obtained data characterize the near-surface area of the system, which at the beginning of the process was the initial solution, and in the end, was an ordered layer of microparticles that are shown in Fig. 2. The change in the properties of the near-surface area occurred in two stages, as indicated by Fig. 3 and equation (13). According to equation (13), processes in the subsurface solution did not affect the flow J of evaporated solvent molecules at t < ti and reduced the flow to the Ji = 0 for t > ti, and the function J(t) underwent the change at t = ti, similar to the changes that take place at the border in supersaturated
0,0 0,5 1,0 1.5 2,0 2,5 3,0 3,5 4,0
z
Fig. 4. Integral distribution functions of the structural elements of the films microrelief on size. 1 - nanoparticles; 2 - nanoparticles with fullerene content 3-10-4; 3 - nanoparticles with fullerene content 1-10-3; 4 - microparticles
metastable environments. Given these facts, we can assume that in the first stage (t < t1), the formation of the nanoparticles and their aggregation onto the microparticles took place, and in the second stage (t > t1), the enlargement of microparticles and their association in the aggregates dominated, forming the nanostructure which is shown in Fig. 2. This conclusion needs to be made to explain the proximity of the 91(l) functions of nanoparticles within the microparticles of different sizes (Figure 4). This closeness indicates that the microparticles were formed surrounded by nanoparticles which have acquired a stable d1(l) function prior to their significant association in the microparticles, i.e. in the first stage of the process. With this in mind, the data in Fig. 4 can be regarded as an appearance of the film's morphological memory, which preserves the information about the two-step process. The presence of morphological memory is also witnessed by formula (15), which allows one to determine the quantity of the fullerene in the system during film formation using the data on the 91 (l) functions. Information on the kinetics of nano- and micro-particle formation can be extracted from formula (14). This formula is a solution of the equation:
d^k(l) _ D d(l) G
~dT _ - Gkl~rn ' (16)
at the boundary condition:
d^k (l)
Gkl^k(l) - Dki^d^ = s(t), (17)
J l^lofc
where Gkl and Dkl are the parameters satisfying the conditions /0 Gkldt = Ak and Dkl/Gkl = pk; 8(t) is the Dirac function. Equation (16) can be obtained using the relations (3)-(6), similar to the equation (7), assuming that Wjk = 0, divJjn = 0, n = nl3/6vAj, and
Gkl = akakl - bkfikl (18)
Dki = 1/2(a| aki + b2k fa
(19)
Here, ak and bk are the effective changes of l at the single acts of integration and disintegration of the particles; aki and 3kl are the repetition frequency of these acts, independent on l, so that:
at Zk = 3ki/aki and q = bk/ak.
These facts allow us to assume that the groups of nanoparticles in the first stage and the microparticles in the second stage were formed according to the same accretion mechanism (i.e. attaching of small particles to larger particles). This is indicated by the fact that the distribution of nanoparticles as well as microparticles is described by equation (14), characterizing a plurality of particles, each of them is coarsened by attaching the fragments significantly smaller than the particle itself. According to relations (14), (15) and (17), the nanoparticle nuclei were in the initial solution or were formed after the start of evaporation during the time much smaller than t1. Their parameter, n10, can be estimated, assuming that n10 = 6130/uAk, where uAk is the volume of the styrene molecule in its crystal. Doing so gives a figure of n10 = (3.1 — 3.2) ■ 108 styrene molecules.
Based on equations (7), (14), (16)-(20), the nanoparticles on the first stage and the microparticles on the second stage undergo accretion independently (Wjk = 0) at homoge-nization of the near-surface solution areas at t > t1(divJjn = 0). This finding is consistent with formula (13). Additionally, relations (7), (14) and (16) indicate that the rate of integration of nano- and microparticles does not depend on their size. According to formulas (3)-(7), the independence of Dkl and Gkl from l may be due to the fact that the nano-and microparticles had constant number of active centers, to which the macromolecules attached. As such centers, the nanoparticles could contain the ends of macromolecule chains, from which they consisted of, collected in the groups responsible for the independence of the parameters ak and bk on l and for the formation of chains of nanoparticles, from which the microparticles were composed.
According to the model ((1)-(7)), the independence of Dkl and Gkl on l does not exclude their dependence on evaporation time. This dependence is reflected by the parameters Ak and pk in equation (14). Equation (14) is a solution of equation (16) for an arbitrary function Ak(t) and a constant parameter pk. According to the formulas (17)-(20), the constancy of the parameter pk may be due to the fact that the frequency functions akl(l,t) and 3kl(-, t) are such that the values Zk and qk are constant or Zkqk << 1. More detailed information on the frequency functions in the model ((3)-(20)) cannot be extracted from the data on microrelief. However, it can be assumed that the function ^1(l) characterizes the primary aggregates of macromolecules (j = 2), as indicated by the value n10, and the function ^2(l) refers to the ordered groups of aggregates (secondary aggregates j = 3). Given this, one can proceed from the function ^1(l) to the distribution '2(n,t)s, using the condition:
where N2 is number of primary aggregates per unit volume of the near-surface layer of the film at t = tF. A similar relation leads to a transition from the function ^2(l) to the distribution '3(n,t)s. At the transition from '3(n,t)s to '3(X,t)s it can be taken that
(20)
(21)
Ps(X,tF ,t ) = 1.
6. Conclusions
It has been established that by evaporation of o-xylene and toluene solutions of polystyrene, polymeric bodies are formed which have a two-tier hierarchical nanostructure microrelief, and each level of the nanostructure has a stable distribution function of its structural elements on size. The results of these functions' determination coincide with the solution of the kinetic equations for the model of polymer body formation by evaporation of the polymer solution, beginning with the primary aggregates of macromolecules (nanoparticles), continued with the association of nanoparticles into the secondary aggregates of macromolecules (microparticles) and completed with the association of microparticles in the polymer bodies.
This model is based on the conservation of the number of atoms in the system and on the consolidation of nano- and microparticles as a result of binary collisions, and does not contain arbitrary assumptions. Therefore, the fit of the model to the experimental data can be considered as an indication of the fact that these processes actually took place during evaporation of the solution of polystyrene, and nano- and microparticles retained the information on how they were being consolidated, and this information can be obtained without destroying the film, defining the distribution function of the structural elements of the microrelief on size. Research methods used in this paper are not specific to the polystyrene. There are a wide range of polymers which have electron micrographs published in the literature that show microrelief, thus making them prime candidates for study using the above-described model.
Acknowledgment
The work was supported by Russia Foundation for Basic Research (grant N 15-43-03034-a.
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