数理解析研究所講究録 1413 巻 2005 年 211-219

211

Dynamic Scaling of the Growing Rough Surfaces
小林奈央樹, 齋藤和亮, 小澤達哉, 森山修, 狐崎創 1, 山崎義弘 2, 松下貢
Naoki KOBAYASHI, Kazuaki Saito, Tatsuya Ozawa, Osamu MORIYAMA, SoKITSUNEZAKI, Yoshihiro YAMAZAKI and Mitsugu MATSUSHITA

Department of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo112-8551
1 Depart ment of Physics, Graduate School of Human Culture, Nara Women’sUniversity, Kita-Uoya-Nishi-machi, Nara 630-8506
2 Department of Physics, School of Science and Engineering, Sh mjuku-ku, Waseda
University, Tokyo 169-8555\S 1. Introduction
The formation of rough surfaces is arich phenomenon in natuoe.l) In astrict
sense, there is almost nothing of asmooth surface but arough one. Typical examviscous flow in porous
ples for the growing rough surfaces include crystal$\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h},2\rangle$.It is one of very important,and bacterial
problems in statistical physics to understand the growing mechanism of such surfaces. In 1985, amily and Vicsek proposed the scaling hypothesis for the growing)
Based on the scaling, alot of experimental and theoretical studroughequa, such as experiments mentioned above and Kardar-Parisi-Zhang
equation, have been carried out to confirm thetion and Edwards-Wilkinson,$\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}^{3)}$mountain$\mathrm{c}\mathrm{o}1\mathrm{o}\mathrm{n}\mathrm{y}^{5)}$$\mathrm{t}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}\mathrm{y}^{4)}$$\mathrm{F}$$\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{s}.6$$(\mathrm{K}\mathrm{P}\mathrm{Z})^{7)}$$\cdot$$\mathrm{i}\mathrm{e}\mathrm{s}$$(\mathrm{E}\mathrm{W})^{8)}$Family-Vicsek scaling behavior.
The Family-Vicsek (FV) scaling is described as follows. Let us consider aroughsurface which grows from the substrate of line seeds. Namely, for simplicity, we takearough surface grown from one dimensional substrate in two dimensional space, i.e.,dimensions. It is straightforward to extend the system to higher dimensions.
In general, the standard deviation $w(L, t)$ of height ffom the substrate is given as$(1+1)$,$\mathrm{f}\mathrm{o}\mathbb{I}\mathrm{o}\mathrm{w}\mathrm{s}$(14)$w(L, t)=$

where

$L$

is the width of astrip of substrate,$t$

the time,$y_{i}$

the surface height at the212

substrate site

$x_{i}$. Here$\overline{h}$is the average height given as$\overline{h}=\frac{1}{L}\sum_{i}^{L}y_{i}$.

$(1\cdot 2)$

Various rough surfaces satisfy the relations$w(L, t)\sim t^{\beta}$, $(t\ll t^{*})$ ,$w(L, t)\sim L^{\alpha}$$(1\cdot 3)$, $(t\gg t^{*})$ ,$(1\cdot 4)$means that the standard deviation $w(L,t)$ grows
where the scalin relationmeans that $w(L,t)$with a power law of time in early stages of growth, whilesaturates to a value at some characteristic time , md then it exhibits a powerlaw of in the late stages. This signifies that the growing rough surface is a selfaffine fractal. The exponents and are called the roughness exponent and growth
exponent, respectively. Family and Vicsek thought that$\mathrm{g}$$(1\cdot 3)$$(1\cdot 4)$$t^{*}$$L$$\beta$$\alpha$$t^{*}\sim L^{z}$,$z= \frac{\alpha}{\beta}$.

$(1\cdot 5)$

is called the dynamic exponent.
AU these relations are unified systematically in the following

where$z$$\mathrm{d}$namic scaling

hypothesis:$w(L, t) \sim L^{\alpha}\Psi(\frac{t}{L^{z}})$,$(1\cdot 6)$

where the scaling fimction
should satisfy the following behavior : $\Psi(x)=$$x^{\beta}(x\ll 1)$ ,$1(x>>1)$ . Using this dynamic scaling hypothesis (the Family-Vicsek$\Psi(x)$dynamic scaling), one can in principle obtain the exponents and . However, it ishard to use the Family-Vicsek dynmic scaling because it is very difficult to whethe values of $w(L, t)$ experimentally and in some case even numerically in early stagesof growth. It follows then that values of the exponent for many growing rough$\beta$$\alpha$$\mathrm{a}\mathrm{i}\mathrm{n}$$\beta$surfaces have not been obtained yet. Any other dynamic scaling approaches areclearly needed to avoid such a difficulty, md obtain reliable values of for familiar
experiments of growing rough surfaces.
$\beta$1.1. Alternative Approach to the $FV$ dynamic $scaling^{10)}$
Our purpose is to examine the scaling exponents and byaltemative
approach to the FV scaling. The fact is, there is the crossover structure at the$\alpha$$\beta$$\mathrm{m}$213

in thetend of , we make use ofplot.
.
oncharacteristic width
FV scaling. Namely, the FV scaling is modified to the following fom of dynamic$L^{*}$$w\mathrm{v}\mathrm{s}$$L$$L^{*}$$t^{*}$

scaling.
$w\sim L^{\alpha}(L\ll L^{*})$,$(1\cdot 7)$$w\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}(L\gg L^{*})$.

Then (8) and (9) are unified into a single dynamic scaling$w \sim t^{\frac{\alpha}{z}}f(\frac{L}{t^{\frac{1}{z}}})$$(1\cdot 8)$$\mathrm{f}\mathrm{o}\mathrm{m}$:,$(1\cdot 9)$$(x\ll 1)$$x^{\alpha}$$f(x)=\{$$(1\cdot 10)$

1$(x\gg 1)$.

Take a few of log-log plots of $w(L, t)$ vs by suitable times to estimate the value of, and then we can examine the exponent by using our dynamic scaling throughdata collapse.
$L$

$z$

$\alpha$Fig. 1. mustration of the rescale of modified FV scaling.

\S 2.

$(1+1)-\mathrm{d}$Eden model||.

well-known exmple. The growth ffie is simple : Let us
For example, we examine the Eden
start ffom the line seed. One of the perimeter sites of cluster is chosen randomly
with equal probability and is incorporated into the cluster as its member, and-known model, it may be the mostprocess is repeated. Since the Eden model isappropriate one to check the validity of our altemative approach. Figure 2 shows
the log-log plots of vs at every interval of $t=20.0$ . The slope yield a value of$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1^{9)}$$\mathrm{t}\underline{\mathrm{h}\mathrm{i}}_{\mathrm{S}}$$\mathrm{w}\mathrm{e}\mathrm{U}$

$w$

$L$

,
. To examine a data collapse, we plot against for values of and .
Figure 3 shows the estimation of data collapse. Our data tend to fah into a single$t^{\frac{\alpha}{z}}$$\alpha$$\alpha\simeq 0.5$,$\frac{L}{-_{\Gamma},tz}$$\alpha$

$z$

214

Fig. 2. The log-log plots of $w(L, t)$ vs$L$

for the surfaces of the Eden model at every interval of$t=20.0$ for System size $=500$ .

Fig. 3. Results of a data collapse for the Eden model with the logarithmic scale. $z=1.5$ ,$\alpha$$=0.5$.

curve for the values of the exponent
and $z\simeq 1.50$ . Hence we obtain thevalues of the exponents that coincide with the-known values of the exponentsof the Eden model. We conclude that the above fact that we can successfully apply
modified dynamic scaling hypothesis to the Eden model assures us of the validity of$\alpha\simeq 0.50$$\mathrm{w}\mathrm{e}\mathrm{u}$it.

215\S 3. Directed Percolation Deppining model
Typical model of paper wetting experiment is Directed Percolation Deppining,(DPD) mode1.11),12) This model is proposed by Buldyrev et. al. and Tang etindependently. The growth rule is following. At first, for au sites we block a fractionof the cells. Let us start h.ne seed. One of nearest neibour site of cluster is chosenrandomly without choosing the block site. To obtain no overhang-interface, weblock sites below the cluster site become wet as well. Noteimpose the rule that
that, on this model, the scaling property is characterized by the directed percolation(DP) problem. Then, for below a critical value the interface propagate withoutstopping. This phase is called moving phase. On the other hand, for above
the interface is pinning by DP cluster, pinning phase. Let us examine the roughness, we take the standard diviation
exponent of DPD model on moving phase. At. Figure 4 shows the$w(L, t)$ as a function of the width of a strip of the substrate
log-log plots of $w(L, t)$ vs for the $p=0.40$ DPD surfaces. Figure 5 shows vs$\mathrm{a}1$$p$$\mathrm{a}\mathbb{I}$$p$$p_{c}$$p$$\mathrm{F}\dot{\mathrm{n}}$$p_{c}$$\mathrm{s}\mathrm{t}$$\alpha$

$L$

$w$

$L$

$\wedge-$$\lrcorner\hat{\yen}$Fig. 4. The log-log plots of $w(L, t)$ vs$p=0.40\simeq p_{\mathrm{c}}=0.47$$L$

for the surfaces of the DPD model for System size $=3000$ .

.

and growthplot. In Figs. 4 and 5, we obtain the roughness exponent
exponent. In Figs. 4 and 5, there are the crossover structure. Instructure, we expect the exisitence of a characteristic length, which changes thenature of interface from $\alpha>0.50$ into $\alpha=0.5$ , i.e. not self-affine interfaces. Next$\alpha\simeq 0.75$$t$$\beta\simeq 0.85$$\mathrm{t}\underline{\mathrm{h}\mathrm{i}}_{\mathrm{S}}$216$\mathrm{o}\mathrm{f}w.(’ L,t)\lrcorner \mathrm{r}_{10}^{\iota}\check{l}\tau 0_{10’}^{\mathrm{z}}$$\mathrm{D}\mathrm{P}\mathrm{D}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1\tau 0^{l}$$\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{t}10^{l}$vs for theFig. 5. The log-log plots$p=0.40,$ cUittiebdowpc$t$

ces of thefor System size $=3000$ .

0.47).

by varying
againstwe try the data collapse by plotting the quantityand . Figure 6 shows the results of the data collapse for DPD model with the. These
logarithmic scale. bom Fig. 6, we obtain the value of $z\simeq 1.0$ and$w/t^{\alpha/z}$$z$$L/t^{\frac{1}{z}}$$\alpha$$\alpha\simeq 0.90$

Fig. 6. Results of a data collapse for the $p=0.40$ DPD model with the lo arithmic scale.
$\alpha=0.90$$z$$=1.0$ ,.

values is different kom the values by “graph” method. This difference is fiom thereason for the crossover structure in Figs. 4 and 5. This stmcture is out of vah.dity

$|$

217

of the FV scaling hypothesis. But at least, owing to very nice data collapse we canconjecture the existence of a kind of dynamic scaling in such interfaces.

\S 4. Multi-affinity for DPD model
If quenched noise leads to power-law distributed noise, so the interface has ahierarchy of local roughness exponent. Just as multifractal, we call it multi-affine
Determining multi-affinity, we measure the gth order correlation function$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}.13)$

defined by the following:$C_{q}(x, t)=\langle|h(x’, t’)-h(x’+x, t’+t)|^{q}\}_{x’,t’}$The scaling exponent$\alpha_{q}$and$\beta_{q}$is defined by the following$C_{q}(x, 0)\sim x^{q\alpha_{q}}$$C_{q}(0, t)\sim t^{q\beta_{q}}$.

$(4\cdot 1)$$\mathrm{r}\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$:,$(4\cdot 2)$.

$(4\cdot 3)$

its interface multi-affine. If theseare dependent of degree , weexponents are independent of , its interface is self-a fEne. For example, we exmine
order correlation function
the DPD model on moving phase. Figure 7 shows theI$\alpha_{q}$or$\mathrm{c}\mathrm{a}\mathbb{I}$$q$$\beta_{q}$$q$$q\mathrm{t}\mathrm{h}$for DPD model. Since the slope on Fig. 7, i.e. aqi is dependent on , the interfaces$q$Fig. 7. The$q\mathrm{t}\mathrm{h}$correlation function for$p=0.40\mathrm{D}\mathrm{P}\mathrm{D}$modelof DPD model are multi-affine. Figure 8 shows the variation withexponent .
$\alpha_{q}$$q$of the roughness

218

07$\mathrm{o}$$0\epsilon$$\mathrm{o}$050$\mathrm{g}’ 0\iota$$\mathrm{Q}$os$0$$\circ$$\circ$$\mathrm{o}$$\mathrm{Q}$$\mathrm{a}$$\mathrm{o}\mathrm{z}$0’02$l$$\iota$70$\mathfrak{g}$42$\mathrm{q}$Fig. 8. The$\alpha_{q}$spectrum for the DPD model.

\S 5. Conclusion
In summary, in order to examine the dynamic exponent , we find an alternative approach to the dynamic scaling hypothesis for the growing rough surface.
-known Eden model to check
We have first investigated the self-affinity of thethe validity of the alternative approach. Next, we have numerically investigated theself-affiity of the DPD model. We have found that the interface for DPD model
was not self-affine, but multi-affine. In future problems, we should investigate thesource of multi-affinity. Barab\’asi and Stanley proposed that power-law distributed. So, we should measure the noise distribution in ordernoise leads toto discuss the Barab\’asi-Stanley suggestion, but it is now under investigation.
$z$$\mathrm{w}\mathrm{e}\mathrm{u}$$\mathrm{m}\mathrm{u}1\mathrm{t}\mathrm{i}- \mathrm{a}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{i}\mathrm{t}\mathrm{y}^{15)}$

References
1) T. Vicsek: Fractal Growth Phenomen\^a 2nd ed. (World Scientific, Singapore,1992).

2) H. Honjo and S. Ohta: Phys. Rev. E49 (1994) 1808.

3) S. V. Buldyrev, A.-L. Barab\’asi, F. Caserta, S. Havlin, H. E. Stanley and T. Vicsek: Phys. Rev. A 45 (1992) 8313.
4) M. Matsushita and S. Ouchi: J. Phys. Soc. Jpn. 58 (1989) 1489.
5) J. Wakita, H. Itoh, T. Matsuyama and M. Matsushita: J. Phys. Soc. Jpn. 66(1997) 67.
6) F. Family and T. Vicsek: J. Phys. A: Math. Gen. 18 (1985) 75

2197) M. Kardar, G. Parisi and Y.-C. Zhang: Phys. Rev. Lett. 56 (1986) 889.
8) S. F. Edwards and D. R. Wilkinson: Proc. Roy. Soc. London A381 (1982) 17.

9) M. Eden: in Proceedeings of the Fourth Berkeley Symposium on Mathematical
Statistics and Probability, ed. J. Neyman ( niversity of California Press, Berkeley,$\mathrm{U}$1961) p. 223.
10) N. Kobayashi, 0. Moriyama, S. Kitsunezaki, Y. Yamazaki and M. Matsushita:J. Phys. Soc. Jpn., 73, (2004) 2112.
. Stanley and T. Vic11). Buldyrev, A.-L. Barab\’asi, F. Caserta, S. Halvin,sek: Phys. Rev. , 45, (1992) R8313.
12) L.-H. Tang and H. Leschhom: Phys. Rev. , 45, (1992) R8309.
13) A.-L. Barab\’asi and T. Vicsek: Phys. Rev. , 44, (1991) 2730.
14) H. Katsuragi and H. Honjo: Phys. Rev. , 67, (2003) 011601.
. Stanley: Fractal Concepts in Surface Growth (Cam15) A.-L. Barab\’asi andbridge University Press, New York, 1995$\mathrm{H}.\mathrm{E}$$\mathrm{S}.\mathrm{V}$$\mathrm{A}$$\mathrm{A}$$\mathrm{A}$$\mathrm{E}$$\mathrm{H}.\mathrm{E}$