Size, shape, growth and reproduction—Towards a physical morphology

Thin Solid Film, 79 (1981) 277-299 PREPARATION AND CHARACTERIZATION

277

SIZE, SHAPE, G R O W T H A N D R E P R O D U C T I O N - - T O W A R D S A PHYSICAL M O R P H O L O G Y GIANFRANCO CEROFOLINI Istituto di Chimica Fisica, Politecnico di Milano, and SGS-A TES, Agrate, Milan (Italy) (Received January 19, 1981 ; accepted February 13, 1981)

The size and shape of real bodies play an important role not only in their aesthetic but also in their physical properties, although they are usually neglected in the physical description. This neglect is irrelevant for bodies with a regular shape in the thermodynamic limit, but is a serious drawback for bodies of small size or of a queer shape. A physical system of volume V is said to be small if V < Vo, where Vo is a characteristic volume associated with the range of the forces acting between the particles, and to be large if V >> V0. If the system is small, shape is always an important feature of the system; it becomes of decreasing relevance for large bodies only if the dimensionality of the system is equal to 3. Physical morphology, the branch of physics which takes into account size and shape, is successful in accounting for several properties of complex systems such as living systems.

1. ROUGHNESSAND CRYSTALSIZE OF FILMS Piero Brambilla, friend more than colleague, proposed to me in 1974, during my stay in Telettra, the problem of relating the "mechanically evaluated roughness" (defined as the average peak-to-valley distance as seen by a moving stylus) to the "atomic surface roughness" (defined as the ratio r of the true surface area At, measured for instance with the standard Brunauer-Emmett-Teller (BET) method1, to the geometric area Ag). This problem had some technological relevance because the roughness factor of ceramic substrates was presumed to influence performance and reliability of thin film circuits. To give a partial solution of the problem, I ignored the non-zero extension of the stylus and I considered the easier problem of finding the roughness factor of a surface with a known profile. This problem is immediately solved with certain welt-known formulae of differential geometry. Thus, if z = z(x, y) denotes the equation of the actual surface for (x,y)~ ~, the true area is given by At = ~¢ (1 d-p 2 +q2)l/2dx dy

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where p = Oz/Ox and q = 8z/Oy. Consequently, the roughness factor is given by

This formula solves the above problem completely but requires a knowledge of the function z = z(x,y). However, this requires a lot of information which is usually unavailable in laboratory practice, in which only the profile z = z(~) along a given direction ~ can actually be measured, and at most the degree of isotropy of the surface can be verified by changing the direction of the test. In the hypothesis of full isotropy, the problem of relating r to an average peakto-valley distance can easily be solved in the case of very rough surfaces. Let It be the arc length in the given direction ~:

= fi{l

(1)

This quantity can be thought of as measured by an ideal zero-diameter stylus running in the interval (0, l). The hypothesis of full isotropy gives (2)

r "~ (IJl) 2

I shall suppose the arc to be formed by a polygon--not only is this hypothesis nonrestrictive but it also adheres more closely to the physical situation. The case of very rough surfaces, i.e. of polygons with angles between adjacent sides very different from ~, can be handled in a simple way. In this case ]z'(~)l ~> 1, at least for ~ belonging to an extended part of the (0,1) segment. However, Iz'(~)l ~> 1 ~ z'(~) 2 ,> 1 and therefore integral (1) can be replaced by

l, =

f f' 0

[z'(~)ld~

0

j"ldzl i Iz,I 0

i=1

where n is the number of sides of the polygon. By definition

laz, i--n i=1

where ~ is the average peak-to-valley distance. If ~- denotes the average projection of each side onto the (0,l) segment, i.e. ~ = l/n, we obtain l, ~ I ~ / S

(3)

When eqn. (3) is inserted into eqn. (2) it follows that 2 r ~ (ff/3)2

(4)

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Although relationship (4) does not completely solve the problem posed by Brambilla, because it does not take into account the non-zero extension of the stylus, it is, however, interesting as it establishes a close correlation between roughness and an overall property of the shape (given by ff and 6). In connection with these computations I was looking for possible applications; in particular, I was trying to determine whether eqn. (4) could be applied to some physically interesting case in the thin film area. While studying the literature I found in Chopra's book 3 that the roughness of metal films deposited onto substrates kept at low temperature increases in proportion to the average number L of layers, i.e. (5)

r = aL

where a is an appropriate constant. A growth in which roughness increases in proportion to the thickness suggests that the surface is very rough. Accordingly, films described by relationship (5) could probably provide a test for the application of eqn. (4). If b denotes the atomic diameter of the metal considered, then ~ = b A L and insertion of this relationship and of eqn. (4) into eqn. (5) gives __

~ a 1/2

AL ~ -~-L

1/2

(6)

This formula shows that the average deviation AL is proportional to the square root of the average thickness L. It is well known that, if X is a random variable described by Poisson statistics, then the standard deviation AX and the average X are related as follows: A~-X= (~)1/2

(7)

The striking analogy between eqns. (6) and (7) suggested to me that eqn. (6) does indeed describe a phenomenon obeying Poisson statistics. In this hypothesis, as soon as the thickness becomes sufficiently great, the Poisson distribution resembles the gaussian distribution; zunder this condition the average deviation AL and the standard deviation AL are nearly equal 4 (AL = (2/r0~/2A"£ = 0.798AT, although in my first pape& on this subject I erroneously confused AL with AL). In these hypotheses eqn. (6) becomes 1~

~1/2 ~- 1/2

If we impose the condition that L is indeed a random variable described by the Poisson statistics, we obtain the relation ~-a]

~1

This formula relates the empirical parameter a to the average projected crystal size 6-: ~ b(2/lta) 1/2

(8)

The average crystal size d-can be estimated approximately by use of the theorem of

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G.F. CEROFOLINI

Pythagoras:

d = (t~2 .+ ~2)1/2 After insertion of eqns. (4), (5) and (8) this gives

The application of this formula to a practical example has led to meaningful results. For copper deposited and kept at - 183 °C, Allen et al. 5 have found that eqn. (5) holds true with a = 0.1, while after annealing the film at 18 °C for 2 h eqn. (5) is still obeyed but a = 0.032. From these values eqn. (8) gives the result that after annealing 6-varies from about 7 to 13 ,~. Simultaneously eqn. (9) predicts that the average crystal size at L = 200 (about 560/~ thick) ranges from 32 to 34 •. Since d-remains almost constant while 6-almost doubles, it can reasonably be deduced that annealing at a moderate temperature does not enlarge the crystals but smooths the angles between the crystal boundaries. As these results are very promising, I proceeded to demonstrate that films deposited onto substrates kept at a low temperature are indeed related to a phenomenon described by Poisson statistics. This demonstration was possible (and easy) with the following hypotheses6: (1) film growth takes places via formation of vertical piles; (2) the probability of multiple condensation on each pile is negligible; (3) re-evaporation is negligible at the deposition temperature; (4) surface diffusion is negligible; (5) condensation kinetics are first order in the impingement rate; (6) all layers are equivalent with respect to vapour condensation. All these conditions are presumably satisfied for the films listed by Chopra. In the following I shall denote the growth occurring according to eqn. (5) as "vertical stack growth". 2. FILM CLASSIFICATION Even having worked for several years in an industrial laboratory institutionally devoted to the study of thin films, I often had difficultyv in defining what is meant by the term "thin films". In microelectronics the classifications thin films and thick films denote two technologies, the former related to vacuum deposition (mainly of metals), photolithographic techniques and so on, and the latter connected with glass/(metal oxide) cermets and with serigraphic techniques. This partition has no physical meaning and I shall not insist in its description. Another classification, of special interest in solid state physics, is that proposed by Aleksandrov s. In this classification thin and thick ~ilms are distinguished according to the ratio of the film thickness bL to a typical length 2 associated with the phenomenon of interest. This length may be the Debye length for charge screening in electrostatic phenomena or the mean free path of carriers in transport phenomena. According to Aleksandrov, a film is said to be thin when bL ~ 2 and thick when bL ~> 2. From a physicochemical point of view, however, the most useful classification is that proposed by Fisher 9. Before describing and commenting on the Fisher classification, I would like to make a few remarks on the meaning of classifying films

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according to their thickness. The phenomena involved in the growth process, from the early stage of less than one monolayer up to the formation of the bulk phase, are of such a complexity that attempts towards their unified description are still immature. Classification is useful to restrict the analysis to limited parts of the whole phenomenology. The overall unsolvable problem is therefore reduced to several easier problems. The solutions to these problems allow the global problem to be reconsidered in a new light and then to be solved, and this solution will probably dissolve the original classification. Fisher classifies films according to their average number of layers in the following way: (discontinuous L ~ 1 purely 2D filmS]thinJ 1< L < 3 2D-to-3D

filmsfintermediate ~thick

bulk or 3D films

3 < L < 30 30 < L < 105-106 105-106 < L

where 2D and 3D mean two dimensional and three dimensional respectively. This classification assumes that the molecular forces which stabilize the film are short range in character (chemical or van der Waals forces), and the borders between various kinds of films are dictated by the following considerations. The definition of a discontinuous film is intuitive and does not require any further comment. The border between purely 2D and 2D-to-3D films, L ~ 3, can be assumed to be given by the smallest thickness for which there is at least a layer whose atoms may have the same coordination number and arrangement as in the bulk phase. As the thickness increases, the number of layers which can have the same coordination and arrangement as in the bulk phase increases; this number is one order of magnitude greater than the number of surface-interface layers when L = 22. Accordingly, L ~ 30 is an appropriate border between intermediate and thick films. The border between thick and bulk films is dictated by results on phase transitions: in experiments on these phenomena a film with L > 105-106 cannot be distinguished from a bulk phase. 3.

FILM GROWTH

The notion of increasing values of L has a clear mathematical meaning but its practical realization depends on the way in which the film actually grows. Depending on the physical and chemical nature of both the substrate and the film, the substrate temperature, the impingement rate and the environment, various kinds of growth can occur. The main modes can be classified as follows lO. (1) Island growth (or Volmer-Weber growth) takes place when the condensing material is much more strongly bound to itself than to the substrate (e.g. metals on insulators). (2) Layer growth (or Frank-Van der Merwe growth) occurs in the opposite situation (e.g. autoepitaxy or materials condensing on more refractory substrates). (3) Island growth after layer growth (or Stranski-Krastanov growth) occurs

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when some layers are required before the m e m o r y of the substrate is lost and the properties of the film material are acquired. (4) Vertical stack growth takes place on substrates at low temperature w h e n the hypotheses listed in Section 1 are satisfied. Modes (1) to (3) can take place even under equilibrium conditions; the analysis of this case has received little or no attention in the literature and this upholds the necessity of the following discussion. A practical way to grow films under equilibrium is to increase the population of a phase physisorbed on a given substrate by increasing the equilibrium pressure. Depending on the relative influence of vertical and horizontal interactions, various situations can occur. Suppose first that lateral interactions determine the phenomenology of adsorption. This does not mean that vertical interaction is negligible--the adphase can exist only because of vertical i n t e r a c t i o n - - b u t that both its strength and its features are almost irrelevant in the physical description of the phenomenon. Suppose that the adsorbent is kept at a temperature below the critical temperature of the gas but sufficiently high to allow adatoms to migrate along the surface. If the surface is very homogeneous and lateral interactions are strong, we run into the situation described so well by Thorny and Duval 11 : we have several (up to five) 2D condensations as the first layers are filled but as soon as L > 4 - 5 no 2D condensation can be recognized and the adatoms behave like a fluid in an external p o t e n t i a l - - t h e adsorption potential. The adsorption isotherm (i.e. the law relating L to pressure at constant temperature) is given by one of the Broekhoff isotherms 12 for L < 4 - 5 and by the Frenkel-Halsey-Hill (FHH) isotherm 13 for L > 4-5. The border between these two descriptions moves toward lower values of L as the homogeneity of the surface decreases. However, in all cases the F H H isotherm can be applied only to films with L > 2-3; at lower thicknesses heterogeneity effects are not negligible and the surface structure must be taken into account. The transition from one behaviour to the other is rather gradual and can be described by associating a new parameter, dimensionality, with any thickness. Dimensionality is a function A(L) which tends to 3 for L ~ + oo and is equal to 2 if the film is purely 2D. A function which exhibits the correct behaviour (A ,,~ 2 for L < 3, A( + oe) = 3, A(L) slowly variable) is A(L) = 2 + L/(L + Lo)

(10)

where Lo is a characteristic thickness of the order of 10. The difference A( + oo) - A(L) = Lo/(L + Lo) is the relative weight of surface to bulk effects: it is the totality for L ~ L o (say L <<.½L o ~ 3) where the film is purely 2D, while it vanishes for L >> L o (say L >~ 3L o ~ 30) where the film becomes thick. If we furnish the imprecise statement a >> b with the precise meaning a 1> 3b (hence a ~ b ¢:~ a ~< ~1b ), the choice L o = 10 centres the zone of transition from A = 2 to A = 3 just in the intermediate region. The F H H description assumes that the adsorbate is liquid-like, so that the kind of growth hitherto considered belongs to the Volmer-Weber mode. Suppose now that adsorption in the first layers is localized because vertical interactions prevail over lateral interactions. In this case, the situation on homogeneous surfaces is described by the BET isotherm 1, in which adsorbed molecules pile up and the structure of the underlying surface is conserved. However,

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as soon as thickness increases, the adsorbed molecules rearrange themselves and eventually acquire the structure of the liquid phase: this process can take place only ifL > 3 and the gradual transition from A = 2 to A = 3 is still described by eqn. (10); correspondingly the isotherm gradually switches from the BET to the F H H isotherm. It is easily realized that this kind of growth belongs to the StranskiK r a s t a n o v mode, and the situation is not modified by heterogeneity effects 14. However, it m a y happen that vertical interactions are very strong, thus allowing the film structure to be very different from the bulk structure even for vapour pressures close to the saturation pressure. This kind of growth belongs to the F r a n k - V a n der Merwe mode, and the BET isotherm applies up to saturation pressure, but as soon as the saturation pressure is reached the vapour can condense on condensation nuclei. Since the film is presumably very rich in nuclei, condensation takes place preferentially on the surface by forming a liquid layer. This phenomenon cannot be described by eqn. (10); an adequate description is instead provided by the conditions

A(L) = 2 A(L) = 3

for L ~< L c for L > L c

(11)

where Lc is a critical thickness that depends on the nature of the surface-film pair. This phenomenon, which can be termed change in dimensionality 15, is a new kind of phase transition. Several cases are known in which the properties of the bulk phase are acquired in a sudden manner; m a n y of these are listed by Van Dongen et al) 6, but the finest example of this kind of phase transition has been described by H o b s o n 17. Table I summarizes the whole situation.

TABLE I Mode

Isotherm in the 2D region"

Isotherm in the intermediate region

Transition to the bulk phase

Equation giving A(L)

Island growth Layer growth

Broekhoff BET

FHH BET

Gradual Sudden (phase transition) Gradual

10 11

Island growth after BET layer growth

FHH

10

"In this region the isotherm depends strongly on surface heterogeneity; in filling the table the surface has been assumed to be sufficiently homogeneous.

For the sake of completeness, I shall finally consider vertical stack growth, although it can never occur under equilibrium conditions. In this case, the film dimensionality remains approximately constant, i.e.

VL: A(L) ~ 2

(12)

because area increases in proportion to thickness even if the film formally extends up to the intermediate and thick region. Relation (5), and hence relation (12), has been shown to hold true up to L ~ 200.

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G . F . CEROFOLINI

4. QUEER SYSTEMS Dimensionality is a physical quantity which can be associated not only with films but also with all large bodies. The definition of dimensionality starts from the remark that the area A of the surface enclosing the volume V of a body cannot be smaller than a given value. Indeed, a well-known variational property of the sphere states that the body of given volume with the smallest area is a sphere. By eliminating the radius R from the elementary formulae V = ~7~R 3 and A = 4~R 2, and remembering the previous variational property, the following inequality is obtained: 1

V <~ - - A

6~a/2

3/2

This constraint limits the possible pairs (A, V) which can be conceived of in our description of nature. The set of all these pairs will be called the euclidean physical world W is. Similarly to the case of three dimensions, any plane surface of given area has a non-vanishing contour and any segment has a non-void border. Hence the volume V, the area A, the length E of edges and the number P of vertices will enter the description of any system. If P v is the bulk atomic density, PA the mean atomic density in a layer of thickness e beneath the surface, PE the mean atomic density in columns of area a beneath the edges and pp the mean atomic density in a volume v beneath each vertex of the body, then the number of corresponding atoms is given by N 1, = pl, PV

(13)

N E = p~(Ea- Pv)

(14)

N a = p a ( A e -- E a )

(15)

N v = pv(V-

(16)

Ae)

(These relationships were given erroneously in ref. 18.) Obviously the sum N = N v + N a + N E + N p is the total number of atoms. Each of these atoms makes its own contribution to the thermodynamic properties; however, in the thermodynamic limit, some of the contributions (13)-(16) may vanish. A system will be said to have dimensionality three (A = 3) when V - - . + oo ~ N ~ N v

dimensionality two (A = 2) when V--.+oo

~ N ~ Nv+ N A

dimensionality one (A = 1) when V - ~ + oo ~ N

~ Nv+Na+N

E

and dimensionality zero (A = 0) when none of contributions (13)-(16) can be neglected. Obviously e is of the order of the range q of interaction, a of the order ofq 2 and v of the order of ¢3. For V ~ ~3 none of contributions (13)-(16) can be ignored;

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however, as soon as V~> ~3 (this is the physical meaning of V~ + co) there is the possibility that only bulk atoms become relevant. The dimensionality is a property of shape such that, as the volume extends, surface, edge and vertex effects can be ignored when A = 3, edge and vertex effects can be ignored when A = 2, only vertex effects can be ignored when A = 1 and no effect can be ignored when A = 0. Thermodynamics is usually concerned with systems for which A = 3 in the limit of very large volumes. Little work has been devoted to the study of small systems (where all the terms (13)-(16) must be taken into account irrespective of shape) or large systems with a queer shape (where at least one contribution in addition to N v must be taken into account even for V ~ + ~). The above definition is taken from a classic work of Yang and Lee 19 where, in order to demonstrate the existence of the thermodynamic limit, "the assumption is made, of course, that the shape of V is not so queer that its surface area increases faster than V 2/3''. The free energy of a given system can always be written as the sum of several contributions: bulk, surface, edge and vertex. For simplicity, I shall limit my attention to bulk and surface contributions, i.e. F =fV+yA

(17)

where f is the (bulk) specific free energy and Vthe surface tension. For large bodies with a regular shape the term vA is always small compared with f V but it can have equally important effects. For instance it is responsible for the equilibrium shape of condensed matter: the spherical form of a liquid drop as well as the Wulff polyhedron of crystalline bodies 2° are due to the small perturbation vA. Since these shapes are regular and this result is obtained only by imposing equilibrium on a body with a free energy that can be described by eqn. (17), we may draw the following important conclusion: the equilibrium shape of a body with only bulk and surface forces cannot be queer. Returning to film growth, it is interesting to note that "queerness" is gradually lost as thickness increases only by films which can grow under equilibrium conditions (according to the Volmer-Weber, Frank Van der Merwe or StranskiKrastanov modes); queerness is never lost by films growing according to the vertical stack mode. For bodies with a regular shape, A increases with the two-thirds power of V, E at most with the one-third power and P remains constant. Thus as soon as V exceeds a certain critical volume (at this point the system will be defined as large) practically only bulk atoms are relevant for the thermodynamic properties: the already quoted result that bodies with a regular shape have dimensionality three follows. For the other bodies, however, A, E or P increase with V faster than the proportionalities mentioned above, so that, even when the system becomes large, surface, edge and vertex effects must be considered. Examples of bodies with a queer shape are films, zeolites and powders. Films have been considered in full detail in the previous sections and for them it is also possible to describe how dimensionality changes with film thickness. Although films form a wide and technologically important class of queer objects, they do not exhaust these systems, as confirmed by the case of zeolites.

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Zeolites, or molecular sieves, are regular lattice structures formed by small crystalline blocks held together in such a way that a complementary lattice of void spaces remains 21. The chemical nature of zeolites is very complex. For example, the chemical composition of zeolite A is 0.23Na20, 0.77CAO, A1203, 1.89SIO2, x H 2 0 ; the water content depends on the conditions of preparation and stabilizes the structure. The structure of zeolite A is shown in Fig. 1 and consists of a framework of negatively charged alumina silicates plus a number of cations sufficient to result in electrical neutrality. The framework is formed by SiO 4 and AIO 4 tetrahedra which leave an almost spherical empty site of diameter 11.4 A (the 0ccavity).

Fig. 1. The structure of zeolite A. The vertices are occupied alternately by silicon and aluminium atoms. Oxygen atoms are situated slightly beyond the edges. The large central cavity, surrounded by eight sodalite cages, is the ct cavity. In the upper right-hand unit, known as a sodalite cage, there is another cavity (the [3 cavity) not considered in the text.

Any increase in the zeolite volume implies that new blocks are added to the system. Since each block increases the area, contour and vertices in proportion to the added volume, zeolites are 3D systems with zero dimensionality (A = 0). In addition to this feature, zeolites are characterized by the following strange topological property: they are connected but not simply connected and the connection multiplicity becomes infinite as V ~ + oo. The strangeness of zeolites exactly matches that of powders: the dimensionality of a powder depends on the sizes of single grains and remains constant as the volume tends to infinity. A relevant example of these systems is given by soil, the dimensionality of which is also a measure of the level of degradation: A ranges from 3 for non-degraded soils to zero for extremely ancient soils. 5. SMALL SYSTEMS

The previous examples of queer systems, with the exception of films, are characterized by being a collection of small systems. To estimate the size at which a system is considered to be small let us consider the body with the "most regular" shape, namely a sphere. For a sphere E = 0, P = 0 and Ae

3e

V - Ae

R - 3e

This ratio is a measure of the relative weight of surface to bulk properties; it is

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negligible when R exceeds 3e by at least one order of magnitude, say when R > 30e. The radius R ~ 30e is an estimate of the smallest radius at which a body becomes large. Since e ~ 3~ this condition becomes R ~ 90~ ~ 250 A. In condensed systems the atomic density is of the order of 1/g 3 and the number of atoms in a sphere of radius 90~ is about 3 x 106. A system with this population is at the border between large and small systems, but it is already macroscopic. In fact, for a system with 3 x 106 particles, statistical fluctuations involve about (3 x 106) 1/2 particles, i.e. 0.5%0 of the whole population. The physical description of small systems requires specific appropriate methods and some theoretical results are known. The most fruitful approach, however, seems to be given by simulation methods of molecular dynamics, which are now possible because of the availability of large fast computers. For small systems the small number of particles assumed to be representative of the system is not a serious constraint. In addition to queer systems, nature abounds in intrinsically small systems, i.e. in systems which cannot be conceptually extended over a given volume. Examples of intrinsically small systems are embryos in phase transitions and living cells. In the first case the properties of the embryo are not the same as those of the bulk, but extension of the embryo invariably leads first to a nucleus and hence to the bulk phase. In the second case the cell is characterized by a proper volume which cannot be extended beyond a given volume above which reproduction takes place. 6. BIOSYSTEMS

Of all the features distinguishing living systems from other systems the most important seems to be the existence of a diffusion field due to metabolic turnover. This field allows matter to flow from the medium to the system, thus permitting the cell to obtain the negative entropy required for the preservation of life function, and in the opposite direction, therefore permitting the cell to emit waste substances. If Q is the reaction rate for the production (Q > 0) or the consumption (Q < 0) of a given substance (Q is given in grams per cubic centimetre per second in the CGS system), D and D are the diffusion coefficients (in square centimetres per second), and c and ~ the concentrations (in grams per cubic centimetre) inside and outside the cell respectively, then the Fick laws and the principle of the conservation of matter lead to the equations DVZ c + Q = c~c/Ot

(18)

DV2~ = c~U~t (19) where V2 is the Laplace operator (V2 = ~2/~x2 + ~2/0y2+ ~2/c~z2) and ~/c3t is the partial derivative with respect to time t. Equations (18) and (19) describe the diffusion in the internal and external media respectively; however, "inside" and "outside" have meaning because a membrane exists which physically separates the cell from the surrounding environment. The main feature of biological membranes is selective permeability--they allow the permeation of metabolic substances (carbohydrates, amino acids, oxygen, water, carbon dioxide etc.), while they prevent the flows of other substances (structural proteins, enzymes, nucleic acids etc.).

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G.F. CEROFOLINI

If no quantity of metabolic substance is produced or destroyed in crossing the membrane and the membrane does not accumulate substance, then we may write - D(Vc" n) a = h(c - c)a - D ( V ~ . n) a =

h(c- C)A

(20) (21)

where h is the permeability of the membrane (in centimetres per second), V the nabla operator, V - O/#R - (~/~3x, ~/Oy, ~/#z), and n the unit vector perpendicular to the surface at the point R and drawn outward; the index A means that all the quantities are evaluated on the membrane. In this work I shall consider only passive transport across the membrane; the finite thickness s of the membrane is simply taken into account by writing (22)

h = ~/s

where @ is the diffusion coefficient of the metabolite inside the membrane. This coefficient does not depend on s and is a characteristic of the substance-membrane pair; obviously the value of ~ can be changed by hormones, antibiotics, acidity etc. I only suppose that h depends on s through eqn. (22). Equations (20) and (21) are coupled boundary conditions to eqns. (18) and (19). The system of equations {(18), (19)} joined with {(20), (21)} has been solved for many systems of biological interest. The exact solution of these equations is strictly related both to the metabolic rate Q(R) and to the cell form assumed; in particular, the spherical form permits simple solutions 22. In order to avoid computation difficulties, I shall consider only the case of uniform Q throughout the cell and of diffusion coefficients D and D sufficiently great to assume the metabolite concentrations to be uniform although not equal to one another. In this case eqns. (18), (19), (20) and (21) must be replaced by Q V - hA(c - ~) = d ( c v )

(23)

hA(c - ~) =

(24)

c V)

The terms d V / d t , d V / d t ( V being the volume of the medium) can be neglected because of the small growth rate of the cell with respect to the rate of concentration variation; thus, on defining z = V/hA

(25)

= V/hA

(26)

eqns. (23) and (24) become dc c - ~ ---t= Q dt

(27)

d~ dt

(28)

c- -

-

0

Since the volume of the medium is usually very large, eqn. (26) gives a very large

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value ofF; in the limit ~ + ~ , eqn. (28) gives E = Eo, a constant. The concentration Eo of anabolites outside the cell is assumed to be high enough to permit all the functions associated with cellular life. Inserting E = E0 in eqn. (27) and assuming Q to be independent of c (see Section 7) we obtain c - Co = zQ{ 1 - exp( - t/z)} + (C - 8o)exp( - t/z) = Q v ~ I - e x p ( - t / ~ + ( C - E0)exp(- t-/

(29)

This formula gives the difference c - E0 as a function of Q, h, V, A, t and of the initial concentration C. For usual values of the parameters affecting z (when not otherwise specified "usual values" are those considered by Rashevsky 22, namely h ,,~ 10 -4 cm s-1, V m 10- 9 cm 3 and A ~ 10- 6 cm 2) eqn. (25) gives r ~ 10 s. Hence, after some 10 s the actual value of c is very close to the steady state value, Eo + Q V/hA, obtained from eqn. (29) for t ~ + ~ . Since the cellular volume does not vary appreciably within some z, the true concentration is always given by c - Eo = Q V/hA

(30)

This result gives the steady state value of c, the concentration being uniform by hypothesis. Obviously, c is not uniform within the cell because the diffusion coefficient is not infinite; however, we are mainly interested in the quantity ( c - E)a which in the steady state is given exactly by eqn. (30), as the principle of the conservation of matter implies. It is worth noting that eqn. (30) assumes passive transport across the membrane and the validity of the theory of ideal solutions. Both assumptions are questionable and surely false in some cases (various "membrane pump" mechanisms have been proposed, especially to account for the N a + / K + ratio z3, and a particular non-ideal water structure has been postulated to account for the same experimental fact without invoking active transport z4' 2 5). However, I am here interested in an overall view, for which eqn. (30) is probably sufficient. To evaluate Q and h we proceed as follows. The Michaelis-Menten 26 theory of enzymatic catalysis postulates that the metabolic reaction of the anabolite M occurs through the action of an enzyme E according to the following scheme: E + M ~ EM

(31)

EM ~ E + products

(32)

If c E is the total concentration of E, free or bound to M in the complex EM, and c is the metabolite concentration, then the rate at which M is destroyed is given by Q - = k cEc K+c

(33)

where k is the kinetic constant of reaction (32) and K is the dissociation constant of complex EM defined by equilibrium (31).

290

G . F . CEROFOLINI

The net rate Q is given by the difference Q + - Q-, where Q + is the rate at which M is produced; Q+ has an analytical expression similar to eqn. (33) because M is produced from the decomposition of a substance M' by the action of an enzyme E'. Two limiting case's can be distinguished: the low concentration limit (c ~ K), for which eqn. (33) gives (34a)

c ~ 0 =~ Q - ..~ kcEc/K

and the high concentration limit (c >> K), for which eqn. (33) gives

C--* + oO ~ Q - --~ kc E

(34b)

The last result is of present interest because the values of the Michaelis constant K range from 10- 5 M to 10- 3 M, while the molar concentrations of the main metabolites exceed 10-3 M. Indeed, glucose in human blood is present at a concentration of about 1~o, corresponding to 6 x l0 -3 M; in mammalian muscle fibre the H C O 3 - concentration is 8 x 10 -3 N 27; and for ATP, ADP and H3PO 4 values such as 10-2 M may be considered near the truth 28. In order to show how the previous considerations can be applied to a metabolic pathway such as eqn. (35), let me first observe that the permeabilities of most metabolites are very small (Ah~ ~ VkiCE,/Ki) so that the concentration c~is limited by the flow across the membrane only if the rate at which M~ is destroyed is smaller than its formation rate. I

lho 1

÷

hN

/

k~ 7

Mo \~¢nk..~

M ~

h1 M1

kN-1]

[ kl

*

(35)

I

i hl

l I shall first consider the opposite case, i.e. the case for which the limiting rate kiCF~ is greater than Qi ÷ . If the molar rate is defined as ql- = Qi-/M~ (where M i is the molecular weight of the substance M~) and if the flow across the membrane is neglected because it is very small, then

qi + --

1 klCE~Ci M i K i 4- c i

291

PHYSICAL MORPHOLOGY

from which it follows that [ kiCE, _11-1

c i = imlqi+

]

Ki

< ( kiCE'/Mi ~)i_lki-~Mi_l

1)-IKI~K i

where I have taken into account that v~_ ~ molecules of M~ are formed from the decomposition of one molecule of M~_~, and that qi+ < vi_~k~_~CE,_,/M~_ ~. It follows that, if the limiting rate at which Mi is destroyed is greater than that at which it is produced, the net formation rate is zero and the concentration is negligible qi = 0

Ci ~ K i

In contrast, if the limiting rate of consumption is lower than the rate of formation, q~÷ > k~CE,/M ~, then the concentration in the cell is limited by the flow across the membrane and becomes much greater than K v In this case the rate at which M i is destroyed is ql- = kiCE,/M! (see eqn. (34b)) and

qi = qi + -- kiCE,/Mi This result holds true independently of the index i. Now, let me apply the above considerations to the substance M~_ 1: the rate at which M~_ ~ is destroyed either equals that at which it is produced (when or is very close to q i - l + < k i _ l C E , _ , / M i _ 1 and thus it is ci_ 1 ~ K i _ l ) k i_ leE,_ ,/M~_ r Tertium non datur. Iterating the previous argument we arrive at the result k c i- 1 klCE~ ~lqv qi = M j 2~j ~' Mii

(36)

where M j (j <~ i - 1) is the molecular weight of the last substance in pathway (35) for which cj ~> Kj. In conclusion, either qi = 0 and c i ~ Ki, or qi is given by eqn. (36) (i.e. it does not depend on any concentration) and c~ is controlled by the flow across the membrane according to eqn. (30). Finally I wish to remark that the previous considerations hold true quite independently of the kinetics assumed--it is sufficient that a condition such as (34) hold true. In particular, the conclusions above can afortiori be applied to allosteric enzyme kinetics too 29. 7.

CELLULAR GROWTH

The mechanism through which a cell grows is at present well known, at least in its essential features: the nuclear DNA produces the complementary DNA and the various kinds of RNA; one of these, the messenger RNA (mRNA), "makes" the proteins (such as enzymes and membrane proteins); the degradation of energy-rich compounds by means of enzymes produces ATP from ADP, and ATP is required in almost all syntheses occurring in the cell. The overall reactions during growth can be summarized as follows: enzymes A ~ RNA ~ proteins ~ membrane proteins !

t structural proteins

292

G . F . CEROFOLINI

At a given point of the growth, through a mechanism at present still unknown, the DNA ceases to produce RNA and begins its duplication. Since mRNA decays with a lifetime much shorter than the duration of mitosis, during this lapse there is practically no protein synthesis. Thus it is possible to distinguish two periods (phases) of cellular life: growth G, during which the cell extends its volume and area, and reproduction R, during which the cell maintains an almost constant volume and eventually changes its area. Actually, the growth is not homogeneous with respect to time. For instance, with respect to the DNA-synthesizing activity, three phases 3° at least 31 should be considered: G1, or gap 1, after the telophase; S, the phase during which the cell synthesizes the DNA which will be used later for the reproduction; and G 2 , o r gap 2, immediately before prophase. However, I shall consider here only homogeneous growth with respect to time, hence neglecting its phases, and I shall look for the laws that relate enzyme concentration and membrane thickness during growth to cell volume and area. To this end I observe that during G the cell produces all kinds of proteins at a rate variable with the protein but always proportional to the concentration of the corresponding mRNA. If any type of mRNA may be considered a constant fraction of the total mRNA, the total quantity ~p of proteins (enzymes, membrane proteins and all other structural proteins), the quantity -~E of enzymes and the quantity ~'m of membrane proteins are given by d.~pp = dt kpCRNAV d~. E

(37)

kEr]ECRNAV

(38)

kmCgNA(m)V ~---kmr]mCRNAV

(39)

dt - - kECRNA(E)V =

d~. m

dt

-

where %NA(X)is the concentration of mRNA devoted to the synthesis of X, ?x is the fraction of mRNA (X) with respect to the total mRNA, kx is the kinetic constant for the synthesis of X, and X stands for E or m. Dividing eqn. (38) by eqn. (37) and eqn. (39) by eqn. (37) we obtain d ~ E = kEr/E d ~ p

kp

d..@m - - kmr/m d.~p kp

which can immediately be integrated (see Appendix A) to give ~ E ---- HE~'P

(40)

~m = Hm~p

(41)

where H E = kg~lE/kpand H m = kmqm/lp. If proteins are present in the cell according to a well-defined ratio ~p with respect to the total cell mass, then

~v = ~vpV

(42)

PHYSICAL MORPHOLOGY

293

where p is the mean density of the cell. Inserting eqn. (42) into eqn. (40) and remembering that c E = .~E/V, we obtain CE = HEC~pp independent of V and A. Similarly, if proteins are contained in the m e m b r a n e according to a well-defined ratio a m, then the membrane volume v is proportional to -~m: V = -~m/~mPm (p,, being the m e m b r a n e density); inserting this result and eqn. (42) into eqn. (41) we obtain (43)

v = mV

where m = Hmo~pp/~mp m

The thickness of the membrane is therefore obtained by dividing eqn. (43) by A : (44)

s = mV/A

All the parameters k x, r/x, p, Pro, ~P, C~m(and hence c E and m) are modified by temperature, acidity, hormones, drugs etc.; however, this fact does not influence the analytical dependence of cE and s on A and V. This dependence plays a fundamental role in further developments. Indeed it allows the functional relationship between c - g and A and V to be found. Insertion of eqn. (44)into eqn. (30) gives Q Vz C--c.=m---_@ A 2

where Q either is zero or is given by eqn. (36) with c~ independent of both A and V. Therefore C--C = zVZ/A 2

where • is independent of A and V. For anabolites ;~ is negative and c < ~. Suppose now the cell has a regular shape. Therefore A oc V 3'2 and c - 6 oc V. Since the difference c - 3 can never be smaller than - ~, V cannot increase indefinitely. Stated another way this means that if growth occurs such that a regular shape is maintained, then as soon as the cell exceeds a given characteristic volume part of the cell can no longer be reached by the anabolites, thus preventing life functions. Conversely, considering anabolites, the positive difference increases in proportion to V. An unlimited increase in osmotic pressure leads to instability which eventually produces cell breakdown. In order to overcome these difficulties, the cell either reproduces, giving rise to two new cells, or grows in a queer shape, e.g. of the kind for which V oc A. 8. REPRODUCTION

Both the possibilities outlined in Section 7 can actually occur, depending on the cell c o n s i d e r e d - - t h e description for bacteria does not apply to eukaryotic cells. Bacteria are monocellular organisms of diameter in the range 1-5 ~tm, while eukaryotic cells are much larger, 10-100 lain. The description of cellular life in terms of surface-volume changes differs appreciably for these systems so that I shall separate my analysis into two parts.

294

G.F. CEROFOLINI

Bacteria are small systems with a diameter of about 1 ~tm and each species of bacteria is endowed with a specific shape. This shape is so characteristic that it allows bacteria to be classified according to their form: we speak of coccus, bacillus, spirillum.., to denote respectively a spherical, ellipsoidal, coiled ... bacterium. The constancy and the regularity of shape during growth imply that the area increases with volume according to the relation V = g A 3/2

(45)

where g is a parameter that is constant with A and V and is characteristic of the bacterium. The phase of growth, probably because of the instability produced by the increase in osmotic pressure, cannot continue indefinitely and at a certain time reproduction must take place. During reproduction the shape must necessarily change because otherwise area and volume should satisfy both the following conditions simultaneously: V 0 = g A o 3/2

2Vo = g(2Ao) 3/2 The first equation states that any newly born bacterium has its own characteristic shape and particularizes eqn. (45) for the first stages of growth. The second equation states that the newly born bacteria occupy the same space as the parent bacterium did immediately before reproduction. These conditions are mutually incompatible so that we must conclude that in a short lapse of time before reproduction (during meiosis) V and A are not related through eqn. (45). The duration of meiosis must be shorter than the growth period because otherwise we would frequently observe bacteria in meiosis; this is not the case and most bacteria actually satisfy condition (45). The simplest way of overcoming this contradiction is to assume that V

= g A 3/2

V = 2V0

for A o ~< A < Af

(growth)

for Af ~ A <7 2A o

(meiosis)

(46)

where the value of Af (the area of the end of growth) is obtained by imposing the condition of continuity, i.e. 2Vo = gAf 3/2. In the {(A, V)} plane the whole process (growth plus meiosis) is represented by a step curve, the first part of which describes the slow phase with constant form (i.e. growth) whilst the second part describes the fast process at constant volume (i.e. meiosis) (Fig. 2). Each of the newly born bacteria grows and reproduces according to eqn. (46) so that after n generations the total volume Vn and area A n are given by

v.

2n x f g A 3/2 [2Vo

for A o ~< A < Af for Af ~ A < 2A o

A n = 2hA The ratio V J A n always remains roughly constant and the whole process is shown graphically in Fig. 3.

295

PHYSICAL MORPHOLOGY

In any generation the total volume is linked to the total area by a function that is always contained between the two straight lines in Fig. 3. The overall process can be seen as a 2D growth. Living systems are characterized by the consumption of energy-rich substances and the simultaneous production of waste materials at a rate proportional to the system v o l u m e I t h e set of these chemical reactions is the metabolism. As the total quantity of substance which can be exchanged by a system with its environment is always proportional to the area, reproduction allows the population of a metabolized system to increase indefinitely. V / A o 3/2

V:

'

1

A3/2

e~n-

/viA// 0.1

,,--.-_,¢_.. growth

0.0 ~ 0.0

/ ,,__.__,x,.__j meiosis

I

l

1.0

2.0

~

A/Ao

Fig. 2. The growth curve o f a l e a Cerofolini, a hypothetical cubic bacterium. During the growth, area and volume are linked by the relation 1

V = --A

6~

3/2

obtained by eliminating the edge length E from the elementary formulae giving area and volume as a function of E. During meiosis V remains constant with A: A o is the area of the bacterium at the start of growth.

The situation is more complex for eukaryotic cells: these systems are larger than bacteria (the diameter of somatic and animal cells usually ranges from 10 to 100.~tm) and are endowed with a richer structure. Membrane thickness is known to be constant (about 100 A) during cellular life; according to eqn. (44) this is possible only if V oc A. A 2D growth, for which area increases in proportion to volume, permits and requires great structural richness at membrane level, with a texture of evaginations and vacuoli deeply extended inside the cell (the endoplasmatic reticulum). This allows the cell to specialize for a given function and prevents the existence of steric constraints to mutual contact. With

296

G . F . CEROFOLINI

mutual contact, building of tissue occurs, and the area available for exchanging matter and energy with the environment is accordingly reduced. V/Ao3/2

/V= 61y~ A 3/2

2.0.

/

/

l

I I

n i it I!

,

1.5-

i

//

t

I I

,/

7

/

/

//

//

•

,/ /

l/

1.0-

/ .

/

.," ./

/-/"

third

generation:

23bacteria

./ /./

,1

//II/~/ / / /"

s /

~

s*

/ O.O

/

second

/J.'" ' //J

generation:

_2 2 ~ bacteria

,.:;.:/..........

0.5.

J

I" J

first

parent

1

generation:2

bacteria

bacterium

~

,

v

~

1

2

4

8

ira.

16 A/Ao

Fig. 3. The growth curve of a spherical bacterium, considered for three generations: the figure shows that

the area and volume remain roughly proportional.

The multicellular organism overcomes this reduction (1) by centralizing the functions of the collection of energy-rich substances and the re-emission of exhausted substances (digestion apparatus) and (2) by distributing the former substances and collecting the latter at high concentrations through a thin texture of tubulations reaching all the cells (circulation apparatus). Thus the topological structure of multicellular organisms resembles that of zeolites. Finally I wish to point out that 2D growth of biosystems is possible only because they are always well away from the equilibrium condition. REFERENCES 1

S. Brunauer, P. H. Emmett and E. Teller, J, Am. Chem. Soc., 60 (1938) 309.

PHYSICAL MORPHOLOGY

2 3 4 5 6 7 8 9 l0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

297

G.F. Cerofolini, Thin Solid Films, 27 (1975) 297. K.L. Chopra, Thin Film Phenomena, McGraw-Hill, New York, 1969. G.F. Cerofolini, Thin Solid Films, 31 0976) L3. J.A. Allen, C. C. Evan and J. W. Mitchell, in C. A. Neugebauer, B. Newkirk and D. A. Vermilyea (eds.), Structure Properties of Thin Films, Wiley, New York, 1959, p. 46. G.F. Cerofolini, Thin Solid Films, 32 (1976) 177. G.F. Cerofolini, G. Ferla and C. Rovere, Thin Solid Films, 50 0978) 73. L.N. Aleksandrov, Phys. Status Solidi A, 44 (1977) I 1. M.E. Fisher, J. Vac. Sci. Technol., 10 (1973) 665. J.A. Venables, Thin Solid Films, 32 (1976) 135. A. Thomy and X. Duval, J. Chim. Phys., 66 (1969) 1966; 67 (1970) 286; 1101. R.H. Van Dongen, Physical adsorption on ionic solids, Thesis, Delft, 1972. W.A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon, Oxford, 1974. G.F. Cerofolini, J. Low Temp. Phys., 6 (1972) 473. G.F. Cerofolini, Z. Phys. Chem. (Leipzig), 258 (1977) 937. R.H. VanDongen, J.H. KaspermaandJ. H. DeBoer, Surf. Sci.,28(1971)237. J.P. Hobson, J. Phys. Chem., 73 (1969) 2720. G.F. Cerofolini, Thin Solid Films, 55 (1978) 293. C.N. Y a n g a n d T . D. Lee, Phys. Rev.,87(1952)404. L.D. Landau and I. M. Lisfshitz, Physique Statistique, Mir, Moscow, 1967, Chap. 15. R.M. Barter, Adv. Chem. Sci., 121 (1973) I. N.I. Rashevsky, Mathematical Biophysics, Vol. l, Dover Publications, New York, 1960. H . H . Ussing, in M. Marois (ed.), Theoretical Physics and Biology, North-Holland, Amsterdam, 1969, p. 179. P.M. Wiggins, J. Theor. Biol., 32 (197 l) 13 I. G.F. Cerofolini and M. Cerofolini, Spec. Sci. Technol., 3 (1980) 315. L. Michaelis and M. L. Menten, Biochem. Z., 49 0913) 333. A. Berkaloff, J. Bourguet, P. Favard and H. Guinnebault, Biologic et Physiologic Cellulaire, Hermann, Paris, 1967. P. C. T. Jones, J. Theor. Biol.,34(1972) l. J. Monod, J. Wyman and J. P. Changeux, J. Mol. Biol., 12 (1965) 88. A. Howard and S. R. Pelc, Heredity, 6 (1953) 261. L.G. Lajtha, C. W. Gilbert, D. D. Porteous and R. Alexanian, Ann. N. Y. Acad. Sci., 113 (1964) 742.

APPENDIX A

Homeostatic effects in growth and reproduction In Section 7 1 considered the problem of finding the ratio ~m/.~V of membrane proteins to total proteins and the ratio ~E/.~p of enzymes to total proteins. These ratios are expressed by the differential relationship d-~x = Hxd-~p

(A1)

where H x is an appropriate constant independent of.~v and X is m or E. In Section 7 1 was interested more in a heuristic foundation of the model rather than in a detailed discussion of mathematical relationships. That context justified the use of the initial condition ~v = 0 ~ - ~ x = 0

(A2)

which permits eqn. (A1) to be solved:

-~x = Hx-~v

(A3)

However, cells have a non-zero lower bound to their size. If-~Vo is the total

298

G . F . CEROFOLINI

protein quantity of a newly born cell then the correct initial condition is -~P = -~Po ~"-~x = -~Xo

instead of condition (A2). The integration of eqn. (A 1) with the correct initial condition gives -~x = Hx(-~P--~Po) +-~Xo

(A4)

Really another index should be used to distinguish the above quantities, since they vary from one generation to the next. Indeed, calling -~xf and .~pf the quantities immediately before reproduction, we may write ~,~pf(n) =

2~po(n+1)

(A5)

.~xf(") = 2 .~Xo("+ 1)

(A6)

where the upper index indicates the cell generation considered. Furthermore I assume Vn: .~po("+ 1)

~--. ,~po(n) = "~Po

(A7)

at least after a sufficient induction time. Inserting eqns. (A5),(A6) and (AT) into eqn. (A4) and considering the first mitosis it follows that -~Xot2) = ½Hx-%o+½-~Xotl) Consideration of the second mitosis gives .~Xo(3) = ~Hx-%o 1 1 ~2 + ~-~Xo 1

1

1

= ~Hx.~po(1 + ~)+ ~--~Xo() and iterating the procedure we obtain 1 ( 1 I ) 1 .~Xo(") = ~Hx.~Po 1 + ~ + . . . + 2 , 2 +2" 1"~x°")

(A8)

The limit of eqn. (A8) for n-~ + oc gives the initial condition after a sufficiently large number of reproduction processes: +~o 1

.~xo(~) = ~1H x'~po,,~o ~ = Hx.~Po

(A9)

Insertion of eqn. (A9) in eqn. (A4) gives eqn. (A3) exactly, therefore justifying condition (A2). The results of this appendix go well beyond the simple demonstrations above and state that the mechanism of growth and reproduction is homeostatic. Indeed, if the value of -~Xois varied by some particular external factor, then after a sufficient number of reproductions newly born cells have no memory of such a perturbation. The only perturbations which can persist are those which modify the value of H x, i.e.

PHYSICALMORPHOLOGY

299

those connected with an altered nuclear activity A1. Correspondingly, if a mutation occurs, then after a sufficient number of cell divisions the memory of the initial character is lost by new cells. Reference to Appendix A

A1 J. Monod, Le Hazardet la Nkcessitd, Seuil, Paris, 1970.