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2
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85034726931
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By deviations we mean smooth, but not necessarily small deformations. So, for example, deformations leading to a sphere are not excluded.
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3
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85034724663
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A lucid discussion of this point, actually much in the spirit of the fractal concept, can be found in Ref. 15, p. 523.
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4
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84952294422
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Referenced as parts I, II, and III.
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8
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85034721446
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Parts of the current work (parts I and II) have been presented at the Fifth European Conference on Surface Science, Gent, 1982
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9
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84952294418
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and at the Israel Chemical Society Annual Meeting, Tel Aviv, 1982.
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11
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84952294421
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and, in 3S, (Technische Universität, Vienna, 1983), p. 233-236.edited by P. Braun, G. Betz, W. Husinsky, E. Söllner, H. Störi, and P. Varga
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(1983)
Symposium on Surface Science
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Avnir, D.1
Farin, D.2
Pfeifer, P.3
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12
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84952294420
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See Ref. 5(a), p. 54-55.
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15
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85034728658
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From a mathematical viewpoint, definition (1) of nonstandard dimension is not the most general one (for example, the limit need not exist). See also Ref. 5(a), pp. and 300. But it is certainly sufficient for all our purposes.
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-
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16
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85034728071
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N(r)
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For example, for a finite number of points, remains constant for sufficiently small r. whence [formula omitted] For a straight line of length L. is the smallest integer [formula omitted] so that [formula omitted]
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N(r)
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17
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85034729476
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See part II. For instance, example 1 mimics anisotropic crystal‐surface roughness. Example 2 anticipates highly porous amorphous solids.
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18
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85034722022
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This shows that, while curves with [formula omitted] substantially [formula omitted] have to be quite irregular, the same value for the surface analog [formula omitted] is attained by comparably mild irregularity if isotropic.
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-
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19
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85034723529
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See Ref. 5(b), p. 145, for a graphic. In Ref. 5(a), p. 164-167, this was called Sierpinski sponge.
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20
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0001247912
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Thus, Mandelbrot’s paradigmatic example of coast lines having [formula omitted] lives in a range of 1-1000 km (Ref. 5). Our surface yardsticks will vary from [formula omitted] to as high as 180 000 [formula omitted] But indirectly inferred values like [formula omitted] for the backbone wiggliness of proteins [ ]may not be easily attributed to a specific such range.
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(1980)
Phys. Rev. Lett
, vol.45
, pp. 1456
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-
Stapelton, H.J.1
Allen, J.P.2
Flynn, C.P.3
Stinson, D.G.4
Kurtz, S.R.5
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27
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85034720863
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AEROSIL (Degussa), CAB‐O‐SIL (Cabot). See also part II.
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28
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85034720976
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Since global deformation of the surface does not affect the asymptotics (2), any other macroscopic shape is equally admissible.
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29
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85034726002
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Back to a single particle, this can also be expressed as “” (Ref. 5(b), Chap. 12): [formula omitted]
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area‐volume relation
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30
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84952276219
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The situation described in Table II is not exhaustive. For example, by scaling up the feth approximant of the Koch surface (Sec. III) by a factor of [formula omitted] one obtains a series of surfaces that are similar within resolution of squares of length [formula omitted] On the k th one of these, Eq. (8) holds [with Eq. (4)] for [formula omitted] and essentially only for this range. Yet, for these surfaces, Table II remains also true if, in columns 4 and 1, [formula omitted] is replaced by any [formula omitted] Thus, when surfaces are not similar within the resolution afforded by the fixed yardstick (here [formula omitted] provided it is sufficiently below [formula omitted] the dependence n vs R may detect a [formula omitted] even with a yardstick below the associated fractal regime. But the absence of such similarity may also prevent the dependence n vs R from detecting the proper [formula omitted] even if the yardstick is in the fractal regime. An example is obtained from assembling [formula omitted] Menger sponges of unit length (Sec. HI) to a “” of length [formula omitted] Within resolution [formula omitted] these cubes are indistinguishable from true cubes and hence similar within this very resolution. On any one of these surfaces, however, Eq. (8) holds with [formula omitted] for [formula omitted] While under any fixed yardstick of radius [formula omitted] Eq. (13) holds with [formula omitted]
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cube
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31
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85034724301
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Menger’s sponge may serve again as a model: When breaking it into pieces, one expects fragmentation along the “” that connect the 8 corner to the 12 bridging cubes. So in the first grinding step, the original sponge of unit length breaks into 20 sponges of length 1/3, each of which is (strictly) similar to the original one, etc.
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faces
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32
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85034722081
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It is instructive to see this for the Koch curve: If the tip of the curve is chosen as origin, no such deformations are necessary. For other choices, some peripheral wings of large‐l segments have to be inverted to achieve similarity.
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-
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34
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85034731058
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For [formula omitted] Eq. (16) is to be read as [formula omitted]
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-
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35
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85034725295
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An alternative proof of Eq. (16) considers channels to and from pores as negligible compared to the cavities themselves (ink‐bottle pores). Thus, while all pores are treated as closed, their walls may be arbitrarily irregular. One then invokes the “” (Ref. 5(b), p. 118-121) to conclude that the number of pores with radius [formula omitted] is proportional to [formula omitted] where D is the total surface’s dimension. Thus, the volume of pores with radius between σ and [formula omitted] is [formula omitted] which is Eq. (16).
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diameter‐number relation
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