arXiv:0911.2497v2 [cond-mat.stat-mech] 11 Jun 2010 i ofiuain hr n hs sams rca ffrac- of fractal mass a is phase geomet- two-phase gen- one a more where in even configuration, exponent an ric the down for write [24] can expression One eral fractals. surface for o betszsfrom sizes object for pta est itiuino tde ytmwti wa a th within of system from studied range transform vector a Fourier of the distribution yields nanom density technique the spatial This at [4–7] th scale. systems for ter fractal techniques investigation of important properties most structural the of one is epeaino h eut eurstertclmodels. theoretical requires results the the experimentally, of available terpretation is space vector wave in range aiu ftesatrn curve scattering the of haviour [22]. methods chemical by obtained also was fracta mer, deterministic the gasket, hexagonal Sierpinski Yamaguchilar [18 by materials out scatter- silica carried porous was neutron self-assembled Small-angle from (SANS) ing [18–20]. developed been system now fractal deterministic obtaining experi for Modern techniques tech- tal structures. the fractal-like ag to of years few preparation due a the probably just is until encountered This limitations nological [15–17]. ha attempts made few made a been only [1 suc- fractals, materials been of deterministic variety have for a However, [8–13] to applied fractals and surface developed cessfully and mass for models ctee 81,23]: [8–13, scatterer α eigageand angle tering where ifatditniyand intensity diffracted are nomto eadn h rca ieso fthe of dimension fractal the regarding information carries ml-nl cteig(A;Xry,lgt etos [1– neutrons) light, X-rays, (SAS; scattering Small-angle ntecs fnndtriitc(adm rcas differ fractals, (random) non-deterministic of case the In h anidctro rca tutr stepwrlwbe- law power the is structure fractal of indicator main The α stepwrlwsatrn exponent. scattering law power the is 2 oi uue ainlIsiueo hsc n ula Eng Nuclear and Physics of Institute National Hulubei Horia atrst ncnrs ihteuulCno e,tefracta the set, Cantor usual the with contrast In set. Cantor ewrs orno rcas atrst AS SAXS SANS, set, Cantor fractals, nonrandom Keywords: 61.05.cf 61.05.fg, ca 61.43.Hv, is numbers: PACS fractal the of gyration f of estimated radius be The can regions. composed Porod is and fractal b the given fr value which the the of of four, particles to generation dimension finite fractal a the for from that vectors shown maxim is and It minima sets. the p but fractal scatterer, a the on of superimposed dimension a maxima fractal which and sets, minima corres represent monodispersive iteration intensities from each considered rules, iterative thre is of to scattering zero set frac from a Cantor and generalized by dimension the generated one from in scattering one small-angle to of zero from vary can ecnie rca ihavral rca ieso,whi dimension, fractal variable a with fractal a consider We λ 10 steicdn aeegh h expnent The wavelength. incident the is .INTRODUCTION I. − α .Y.Cherny, Yu. A. 10 4 = to I q ( to D 10 q = ) 10 o asfatl and fractals mass for − 1 ∼ on nttt o ula eerh un 490 Russia 141980, Dubna Research, Nuclear for Institute Joint 4 1 λ 4 tal. et π A q 13.Sneol finite a only Since [1–3]. A ˚ ˚ − sin − α 1 cteigfo eeaie atrfractals Cantor generalized from Scattering , n stu eyeffective very thus is and θ 1 20)[1.Amolecu- A [21]. (2006) where , .M Anitas, M. E. 2 θ α stescat- the is Dtd coe 3 2019) 23, October (Dated: 6 = ,2 1, I poly- l have s sthe is .I Kuklin, I. A. men- − in o ent (1) in- 4]. ve ve D 3] e- e e ] odn oacranfatlgnrto.Small-angle generation. fractal certain a to ponding a dimension tal retdgnrlzdCno esi bandanalytically randoml obtained of is intensity sets scattering dimensio Cantor The fractal generalized the oriented factor. control scaling se to Cantor a us of using allows generalization a that paper, considered this is In model systems. the real applying to on restrictions some imposes which sion, distribu- Schulz a over fr [25]. averaging function similar by tion a sponge, for Menger obtained the results functi tal, the distribution with the consistent of is width This oscillat increasing local with these out account, cancel into non- taken of polydis- polydispersity is the disorder, fractals when random physical of that real show kind hand, we specific Below, other persity. a the the exhibit On follows always which (1). systems equation of th of behaviour in law the as power out, fractals, dis smeared random is are there of singularities If case the then derivatives. system its the or in s function of this reason in by oscillates larities function, distribution pair the of rlzdpwrlw hsi u osailodri non-rando in gen- curve order scattering a spatial the called to fractals: due be is can This and which law. law, maxima power eralized power of a superposition surface-lik as successive decaying other minima a or [15] shows jack [17], fractal type s sponge, fractals, Menger non-random the from as scattering the (1)], [equation part. remaining th each and for rule repeated iterative is an to process according removed then (iter are subparts them generator into of a divided is and initiator an (initiator) Usually, set operation). initial an of presence nadto,tebudr ufc ewe h hssalso phases dimension the of between fractal surface a boundary forms the addition, In ufc fractal surface α rca,w have we fractal, ca,teepnn hne tsfcetylrewave large sufficiently at changes exponent the actal, a ntredmnin scluae.Tesse is system The calculated. is dimensions three in tal h sa oo a.I ssonta h ubrof number the that shown is It law. Porod usual the y and factor, scaling a using controlled is dimension l cltdanalytically. lculated 2( = r apdwt nraigpldsest fthe of polydispersity increasing with damped are a neig O072 uhrs-auee Romania Bucharest-Magurele, RO-077125 ineering, ernol retdadpae.Tescattering The placed. and oriented randomly re h xsigmdl 1,1]asm ie rca dimen- fractal given a assume 17] [15, models existing The ncnrs ihtesml oe a eaiu above behaviour law power simple the with contrast In the assumes models fractal non-random of construction The o h au ftebudr ewe h fractal the between boundary the of value the rom 1 hi eeaiaino h elkontriadic known well the of generalization a is ch wrlwdcy ihteepnn qa othe to equal exponent the with decay, law ower .Balasoiu, M. D ntredmnin.Teitniyprofile intensity The dimensions. three in e m + D D D p m m D ) ,2 1, otiigprso rca dimension fractal of pores containing − = s Federation n = D n .A Osipov A. V. and D s p D − 3 = m 6 I ntepriua aeo mass a of case particular the In . < ( and D q ) s 3 en h ore transform Fourier the being , h xoette reads then exponent The . 2 and D < 1 D s p < 3 = 3 . hl o a for while , Some . order ingu- ative ions D uch on. ac- for m p ts n y e e e s . 2 mono- and polydisperse systems. Our method is based on the approach that was successfully employed in some aspects for other similar systems of non-random fractals such as the Menger sponge [15]. Numerous examples of the fractal-like behaviour were ob- tained experimentally in collaboration with one of the authors of this paper (AIK) for different kind of objects, namely soils [26–28], biological objects [29] and nanocomposites [30]. Only three parameters are extracted from the fractal scatter- ing intensity: its exponent α [see equation (1)], and the edges of the fractal region in q-space, which appear as ”knees” in the scattering line on a logarithmic scale. Other parameters are not usually obtained from small-angle neutron scattering (SANS) curves. If some features of the fractal structure are available from other considerations, one can construct a model and obtain ad- FIG. 1: Zero, first and second iterations (approximants) for the ditional information on the fractal parameters. Note that a real generalized Cantor set, composed of cubes (upper panel) and balls physical system does not possess an infinite scaling effect and (lower panel). the number of iterations is always finite. In this paper we cal- culate the scattering from generalized Cantor sets for a finite number of iterations. As discussed below, many features of the scattering are quite general; in particular, one can estimate the number of particles from which the fractal is formed. FIG. 2: The first five iterations for the one-dimensional generalized Cantor set with scaling factors γ = 0.2 (left) and γ = 0.5 (right). It II. CONSTRUCTION OF THE GENERALIZED CANTOR is generated by removing at the mth iteration the central interval of m SET length γ l0 from each remaining segment. The fractal (Hausdorff) dimension of the one-dimensional generalized Cantor set is equal to − ln 2/ ln βs, where βs is given by equation (4). At γ = 1/3 The well known Cantor set is created by repeatedly delet- we obtain the usual Cantor set with fractal dimension ln 2/ ln3 = ing the open central third of a set of line segments. One can 0.6309 . . .. construct various generalizations of the set by, say, cutting the segments into a different number of intervals or choosing an- other scaling factor (see, e.g. [31]). Below, we explicitly de- The Hausdorff dimension [5] of the set can be determined D scribe a three-dimensional generalization of the Cantor set, from the intuitively apparent relation Nm (l0/lm) for a which is used for studying the scattering in the subsequent large number of iterations, which yields ∼ sections. ln N 3 ln 2 The generalized Cantor set in three dimensions can be con- m (3) D = lim→∞ = , structed from a homogeneous cube by iterations, which we m ln(l0/lm) − ln βs will call approximants. The zero-order iteration (initiator) is where the scaling factor 0 < β < 1 for each iteration is the cube itself with side length , which can be specified in s l0 defined as Cartesian coordinates as a set of points obeying the conditions , , . l0/2 6 x 6 l0/2 l0/2 6 y 6 l0/2 l0/2 6 z 6 l0/2 βs (1 γ)/2. (4) The− first iteration removes− all points from− the cube with co- ≡ − ordinates γl /2 C. Discussion of results 0 10 These results are in accordance with scattering from sim- m=2 ilar systems like the Menger sponge, displaying the same -2 m=3 behaviour of the scattering curve [15]. Those authors first 10 omitted the last multiplier in equation (15), assuming that m=1 I m m ql0βs 1, and then used the expansion of cosines that allows for≪ reducing the intensity formula to a sum of quite -4 Cubes 10 simple terms. However, the number of terms grows exponen- D=2.27 ( =0.2) tially on increasing the number of iterations, and the method becomes very time-consuming even for m = 4. By con- trast, using the straightforward integration in equation (18), -6 10 we can employ the exact relation [equation (15)] for arbi- 1 10 100 trary m, which can yield a substantial improvment in accu- (a) ql 0 racy. At sufficiently large ql0, the main contribution to the in- tegral comes from a small number of narrow and high spikes, 0 10 which is typical for an interference pattern. Nevertheless, the integral can be evaluated even at sufficiently large values of m=2 4 ql0,up to 10 in a reasonable amount of time. m=3 -2 10 I 1. The fractal structure factor m Balls, R=l /2 0 D=2.27 ( =0.2) -4 10 The scattering intensity can be approximately represented as m 2 m=1 Im(ql0) Sm(q) F0(βs q) /Nm, (19) -6 10 ≃ h| | i where 1 10 100 (b) ql 0 S (q) N P (q) 2 . (20) m ≡ mh| m | i FIG. 3: Scattering intensities [equation (18)] for the first three iter- m ations of the generalized Cantor set composed of (a) cubes and (b) Here, Nm = 8 is the total number of cubes (balls) in the balls. The radius of the initial ball (initiator) is R = l0/2. The in- mth iteration, and the quantity Sm(q) is the fractal structure tensities can be estimated as a product [equation (19)] of the fractal factor (see, e.g. [33]). The latter is Nm times larger than the structure factor and scaled zero iteration, shown in Fig. 4. intensity given by equation (18) with F0 = 1, for instance when the radius of the ball tends to zero. The form factor of the ball [equation (9)] is isotropic, and hence relation (19) where is satisfied exactly. The structure factor carries information about the relative positions of the cubes (balls) in the fractal. Pm(q) G1(q)G2(q) ...Gm(q), (16) ≡ Indeed, using the definition (6) and the analytical formula for and the form factor [equation (15)], one can obtain m−1 m−1 m−1 Gm(q) cos(qxl0βtβ )cos(qyl0βtβ )cos(qzl0βtβ ) 1 ≡ s s s (m) (17) Pm(q)= exp[ irj q], (21) Nm − · 16j6Nm for m = 1, 2,.... We can also put, by definition, G0(q)=1 X and P (q)=1 in order to describe the initiator. 0 (m) To obtain the cross section given by equation (5), we should where rj are the center-of-mass coordinates of cubes 2 calculate the mean value of Fm(q) over all directions n of (balls). Then the fractal structure factor [equation (20)] reads the momentum transfer using| equation| (7) (m) 2 sin qr q 2 jk (22) Im(ql0) Fm( ) , (18) Sm(q)=1+ (m) , ≡ h| | i Nm 16k out of phase. This happens when the most common distance 0 10 between the center of mass of the objects equals π/q. Taking k−1 the value of the distance 2l0βtβs (see Sec. II), we obtain an m=0 estimation of the minima positions for the mth iteration -4 I q π -2 10 q l , k =1, . . . , m. (26) Cube k 0 k−1 ≃ 2βtβs I m One can see that the structure factor has as many minima as the number of iterations. -4 Ball, R=l /2 10 0 2. Generalized power law 1 10 As was noted in Sec. I, rigid spatial correlations between (a) ql 0 points in a non-random fractal yield singularities in the form of δ-functions, which leads to oscillations in the structure fac- 0 10 tor (Fig. 4b). The weight of each δ-functions follows the (m) m=1 power law with respect to the distance between points rjk , as determined by the fractal dimension. As a consequence, the m=2 scattering curve shows groups of maxima and minima which -2 10 are superimposed on a monotonically decreasing curve pro- m=3 portional to some power of the absolute value of the scatter- S / N m m ing vector (generalized power law). Short-range correlations govern the long-range wave vector oscillations and vice versa. -4 10 It is essential that the oscillations are not damped on increas- ing the number of iterations, because this influences only the D=2.27 ( =0.2) short-range correlations and hence the scattering behaviour at m=4 large momentum. -6 An advantage of the model considered here is the explicit 10 1 10 100 analytical expressions (15)-(17), which allow us to estimate (b) ql 0 easily the fractal region where the generalized power law aplies. If the cosine argument in Gm+1 is much smaller than FIG. 4: Scattering intensities [equation (18)] for various system ge- 1, then a further increasing in m does not lead to an essential ometries. (a) Zero iteration: intensities for a cube of edge l0 and a correction and the mth iteration reproduces the intensity that ball of radius R = l0/2. The positions of the maxima obey Porod‘s law. (b) The structure factor of the generalized Cantor fractal [equa- the ideal Cantor fractal would give at this argument. Thus, the m th iteration works well within the region m , where tion (20)] in units of Nm, where Nm = 8 is the total number of m ql0βs 1 we use the estimation β 1. Such behaviour can≪ be seen in cubes (balls) in the mth iteration. The diagram represents the first t ∼ four iterations at γ = 0.2, which corresponds to the fractal dimen- Fig. 3. sion D ≃ 2.27. The asymptote for the structure factor equals 1 when On the other hand, within the Gunier region ql0 . 1 the m ql0 & 1/βs (see text). The deep minima marked by vertical arrows intensity is very close to 1 and we have a plateau in the log- can be estimated using equation (26). arithmic scale. The generalized power-law behaviour is then observed in the region m At zero momentum, relation (22) reads S (0) = N . At 1 ql0 1/β . (27) m m ≪ ≪ s sufficiently large momentum, equation (22) yields the asymp- This estimation indicates two typical length scales important totic relation for a fractal, its size l0 and the characteristic distance inside the th iteration m [17]. In the fractal region mq Sm(q) 1. (24) m l0βs F0(βs ) ≃ 1, and we obtain from equation (19) ≃ The underlying physics is quite clear: when the reciprocal I (ql ) S (q)/8m. (28) wave vector is much smaller than the characteristic variance m 0 ≃ m of the distance between the points then the scattering pattern Beyond the fractal region, when the inequality of equa- does not “feel” their spatial correlations. The characteristic tion (25) is satisfied, we derive from equations (19) and (24) m m variance is of the order ql0βtβ ql0β , and the asymptote s ∼ s m 2 m is attained when Im(ql0) F0(βs q) /8 . (29) ≃ h| | i m It follows that, beyond the fractal region, the scattering inten- ql0 1/β . (25) ≫ s sity resembles the intensity of the initiator, i.e. a cube or ball The structure factor reaches its deepest minima (see in the present case (Fig. 5). In particular, the maxima of the Fig. 4b) when most of the objects inside the fractal interfere curve obey Porod’s law [3]. 6 0 10 3. The radius of gyration =0.25 r One can calculate the radius of gyration Rg of the frac- m -2 =0.4 S /8 10 r m tal. By definition [1], in the Guinier region I(q) = I(0)(1 2 2 − q Rg/3+ ...). Expanding equation (15) in a power series in I m -2.27 q and substituting the result into equation (18) yield I q -4 10 2m Cubes 2m 2 2 1 βs 2 Rg = βs Rg0 +3βt − 2 l0, (30) D=2.27 ( =0.2) s 1 βs -6 − 10 -4 m=4 I q where Rg0 = l0/2 for a uniform cube and Rg0 = R 3/5 for =0, Cubes r a uniform ball. The limit m gives the radius of gyration -8 → ∞ p 10 of the ideal Cantor fractal 1 10 100 1000 (a) ql 0 √ 3βtl0 0 R = . (31) 10 g 2 1 βs =0.25 − r As expected, the characteristicsp of the ideal fractal do not de- -2 m 10 pend on specific shapes from which the fractal is constructed. S /8 m -2.27 I I q m -4 =0.35 IV. FORM FACTOR OF POLYDISPERSE SETS 10 r Here we study a specific kind of polydispersity, when the Balls, R=l /2 0 -6 whole fractal is proportionally scaled and the scales are dis- 10 D=2.27 ( =0.2) -4 tributed. This implies that it is sufficient to consider the dis- I q m=4 tribution of the fractal length in the above formulae. Then the =0, Balls r distribution function DN (l) of the scatterer sizes can be con- -8 10 sidered in such a way that DN (l)dl gives the probability of 1 10 100 1000 (b) ql finding a fractal whose size falls within the interval (l,l + dl). 0 0 The intensity can then be written as 10 ∞ =0.25 2 2 r poly 0 Fm(q) Vm(l)DN (l) dl I (q)= h ∞ i , (32) m 2 -2 V (l)D (l) dl m m N 10 R 0 S /8 m -2.27 I q where V is the total volumeR of the mth approximant, I m m -4 =0.35 3 3m 10 r V = l (2β ) . (33) m s The influence of particle size distribution on the intensity Balls, R=l /4 0 -6 scattering curves is essentially determined by its breadth. Nar- -4 10 D=2.27 ( =0.2) I q row distributions (i.e. log-normal, Schulz, Gauss) have no in- m=4 fluence on the scattering exponent but control a smoothing of =0, Balls r the intensity curve, reproducing the power law behaviour for -8 10 a certain value of the variance, while broad ones can change 1 10 100 1000 (c) ql 0 even the scattering exponent [12]. In order to study this influence on scattering from the Can- FIG. 5: Influence of polydispersity on scattering from Cantor fractals tor sets, we consider a log-normal distribution given by of (a) cubes and (b) and (c) balls. The radius of the initial ball (ini- = 2 = 4 2 2 tiator) is R l0/ in part (b) and R l0/ in part (c). The upper 1 ln(l/l0)+ σ /2 monodispersive curve is the structure factor [equation (20)], which DN (l)= exp , (34) √ 2σ2 in the fractal region [equation (27)] is proportional to the exact inten- σl 2π "− # sity [equation (28)]. The borders of the fractal region are shown with σ ln(1 + σ2). filled (violet) triangles on the horizontal axes. In this region, the ex- ≡ r ponent of the polydisperse curve equals the fractal dimension, while Here, l and σpare the mean length and its relative variance, ≫ 1 m 0 r at large momentum qR /βs , the power law exponent changes respectively, i.e. from the fractal exponent to four, given by the usual Porod law. The beginning of the Porod region is shown by the filled (violet) squares. l l , σ l2 l2/l , (35) 0 ≡ h iD r ≡ h iD − 0 0 q 7 ∞ where ... D 0 ...DN (l) dl. One can see that the rela- controlling the fractal dimension, which can be varied from tive varianceh i regulates≡ the width of the distribution. zero to three. This is beyond the standard for mass fractals Fig. 5 representsR the scattering curves from mono- and from an experimental point of view [14], since fractal di- polydisperse Cantor sets for the fourth iteration at γ = 0.2. mensions less than 1 have not yet been observed. The form The curves corresponding to the polydisperse cases approach factor of the generalized Cantor set is calculated analytically the monodisperse ones as the distribution width σr tends to for arbitrary iteration. This allows us to evaluate the scatter- zero. Smoothing of intensity curve [equation (32)] increases ing intensity for mono- and polydisperse fractals by means of when the width of the distribution σr becomes larger. The simple integrals. We find the values of the asymptotes of the most interesting effect is changing the power law exponent fractal structure factor and reveal their nature. The radius of from the fractal dimension D in the fractal region [equa- gyration is obtained analytically as well. tion (27)] to the usual Porod exponent of four beyond this re- The suggested model describes changing the power law ex- gion. Here the intensity is approximately proportional to the ponent from the fractal dimension to the usual Porod exponent scaled intensity of the initiator, according to equation (29), beyond the fractal region. In the intermediate region a typical thus giving Porod’s law. well or “shelf” arises, which can be observed experimentally One can see the well in the intermediate region ql0 [29]. In comparison with Schmidt and Dacai (1986) [15], we m ∼ 1/βs , where the monodisperse curve is shifted up to the not only calculate the scattering intensity for large momenta m asymptote 1/8 . The well becomes even more pronounced but also explain its behaviour. Moreover, if an experiment ob- when the size of the initiator becomes smaller, compare serves the threshold between the fractal and Porod regions, the Fig. 5(b) and (c). For a very small initiator size, a plateau approach suggested here yields the number of particles in the should appear here. The reason is that the border of the Porod fractal set. Another way to describe the experimental data for region shifts forward, and until it reaches this border the in- fractals is ab initio methods, see e.g. [35]. tensity profile coincides with the smoothed structure factor, In a number of cases we have a priori information about which has an “infinite” plateau. Such behaviour of the scat- the fractal structure, for instance when the fractal synthesis tering intensity has been observed experimentally [29]. A is well controlled by chemical methods. One can then build phenomenological approach has been developed by Beaucage a fractal model with many parameters and use this model to (1995) [34] for studying various exponents in the scattering obtain additional information from the scattering data. The intensity. generalized Cantor set is an example of such a construction. Note that the appearance of the well or “shelf” between the A similar scheme can be developed by the same method for fractal and Porod regions near the volume 1/Nm is a con- other types of fractal sets. sequence of the general asymptote 1 of the fractal structure factor [equation (24)]. Thus, the position of the “shelf” allows The model can be extended to match specific needs in var- us to estimate the volume of particles that form the fractal, ious ways, including different values of the scaling factor at provided the total fractal volume is known (see discussion in each iteration, and different probability distributions for the Sec. IIIA). fractal length and the shapes from which it is constructed. Expanding equation (32) in a power series in q and inte- grating over the distribution, we arrive at the radius of gyra- tion given by equations (30) or (31) multiplied by the factor 2 13/2 Acknowledgments (1 + σr ) . The authors are grateful to A. N. Ozerin and V. I. Gordeliy V. CONCLUSIONS for fruitful discussions. The work was supported by Rus- sian state contract No. 02.740.11.0542, a grant of Roma- We consider a model that generalizes the well known one- nian Plenipotentiary Representative at JINR, and the projects dimensional Cantor set. 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