# Critical Sample Aspect Ratio and Magnetic Field Dependence for Antiskyrmion
Formation in Mn1.4PtSn Single-Crystals

B. E. Zuniga Cespedes  Institute of Applied Physics, Technische Universität
Dresden, D-01069 Dresden, Germany  Max Planck Institute for Chemical Physics
of Solids, D-01187 Dresden, Germany  P. Vir  Max Planck Institute for Chemical
Physics of Solids, D-01187 Dresden, Germany  P. Milde  Institute of Applied
Physics, Technische Universität Dresden, D-01069 Dresden, Germany  C. Felser
Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany
ct.qmat, Dresden-Würzburg Cluster of Excellence-EXC 2147, TU Dresden, 01062
Dresden, Germany  L. M. Eng  Institute of Applied Physics, Technische
Universität Dresden, D-01069 Dresden, Germany  ct.qmat, Dresden-Würzburg
Cluster of Excellence-EXC 2147, TU Dresden, 01062 Dresden, Germany

###### Abstract

Mn1.4PtSn is the first material in which antiskyrmions have been observed in
ultra-thin single crystalline specimens. While bulk crystals exhibit fractal
patterns of purely ferromagnetic domain ordering at room temperature, ultra-
thin Mn1.4PtSn lamellae clearly show antiskyrmion lattices with lattice
spacings up to several µm. In the work presented here, we systematically
investigate the thickness region from 400 nm to 10 µm using $100\times 100$
µm2-wide Mn1.4PtSn plates, and identify the critical thickness-to-width aspect
ratio $\alpha_{0}=0.044$ for the ferromagnetic fractal domain to the non-
collinear texture phase transition. Additionally, we also explore these non-
collinear magnetic textures below the critical aspect ratio $\alpha_{0}$ above
and below the spin-reorientation transition temperature $T$SR while applying
variable external magnetic fields. What we find is a strong hysteresis for the
occurrence of an antiskyrmion lattice, since the antiskyrmions preferentially
nucleate by pinching them off from helical stripes in the transition to the
field polarized state.

##  I Introduction

Magnetic materials that exhibit topological spin arrangements are promising
candidates for future applications, in particular for spintronics. One type of
such topological arrangements are the so–called skyrmion lattices (SkLs) that
may arise in the presence of an antisymmetric Dzyaloshinskii-Moriya
interaction (DMI). Every skyrmion represents a spatial distribution of non-
coplanarly arranged magnetic moments, whose orientations once cover all
spatial directions. This mutual noncoplanar orientation of neighboring spins
is generally described by the topological charge that, in the end, is able to
differentiate between skyrmions (Skys) and antiskyrmions (A-Skys) [1, 2, 3, 4,
5, 6].

To date, a large number of skyrmion-hosting compounds has been discovered,
such as the B20-type chiral magnets [7, 8, 9, 10, 11], $\beta$-Mn-type Co-Zn-
Mn alloys [12], Cu2OSeO3 [13, 14, 15, 16], or some lacunar spinels [17, 18,
19], which all support Bloch- or Néel-type skyrmions, respectively. So far,
antiskyrmions were observed in thin-plates of the tetragonal Heusler compounds
Mn1.4Pt0.9Pd0.1Sn[20, 21, 22, 23], Mn1.4PtSn [24], Mn1.3Pt1.0Pd0.1Sn [25] and
Mn2Rh0.95Ir0.05Sn [26], only. These first Lorentz transmission electron
microscopy (LTEM) measurements demonstrated the nucleation of a triangular
antiskyrmion lattice (ASkL) in a magnetic field that was applied both
perpendicular to the surface of a thin lamella sample, and parallel to the
tetragonal $c$-axis of these crystals [20, 21].

Subsequent LTEM experiments revealed that for some particular temperature and
field regions within the phase diagram as well as for selected sample
thicknesses, A-Skys may also arrange in a squared lattice fashion [22, 23].
Moreover, elliptically-distorted skyrmions of both handedness as well as the
non-topological bubble lattice were shown to appear when applying the
symmetry-breaking in-plane magnetic field in combination with an out-of-plane
field [22, 23]. This makes the tetragonal Heusler compounds a unique class of
materials hosting a rich variety of controllable magnetic textures.

Moreover, in bulk samples of Mn1.4PtSn an anisotropic fractal ferromagnetic
closure domain pattern (FMD) has been reported in clear contrast to the LTEM
results, which shows, that Mn1.4PtSn is an easy-axis ferromagnet at room
temperature with the easy-axis aligned along the tetragonal $c$-axis and a
pronounced quadratic anisotropy within the $ab$-plane, and highlights the
importance of long-ranged dipolar magnetic interactions for the formation of
antiskyrmions in these materials [27]. Also, a recent study of the magnetic
textures in Mn1.4PtSn using wedge-shaped samples came to the same conclusion
[24]. Furthermore, at $T$SR$\sim$170K Mn1.4PtSn undergoes a spin reorientation
transition from the coplanar above $T$SR to the non-coplanar spin orientation
below $T$SR [28, 29].

Hence, to clearly determine the subtle differences in the phase diagram as
well as to safely quantify the crossover from the reported fractal
ferromagnetic domain (FMD) to the non-coplanar texture (NCT) phase in
Mn1.4PtSn as needed for instance for realizing miniaturized room-temperature
devices based on A-Skys, it is essential to characterize the different
magnetic structures over a broad thickness range on samples with no or only
weak thickness variations on the length scale of the antiskyrmion radius. We
thus employ in the present study magnetic force microscopy (MFM) in order to
clearly resolve the different magnetic textures of Mn1.4PtSn in Mn1.4PtSn
plate samples cut out from bulk single crystals by means of focused ion beam
(FIB) milling. What we observe is how the different magnetic patterns depend
on sample thickness, and how and when these structures change when varying
both temperature and the applied magnetic field.

Magnetic force microscopy (MFM) has proven to be a valuable tool when studying
complex spin textures such as helices and skyrmions as well as complex domain
patterns [11, 16, 17, 30, 31]. MFM can be effectuated with high precision not
only under the various temperature and magnetic field conditions, but
moreover, can be applied to investigate near-surface properties of any sample
thickness, from monolayer to bulk systems. In this sense, MFM is not
restricted to the analysis of ultra-thin lamellae, but will provide insight
for any of these Mn1.4PtSn plate samples.

The paper is organized as follows: After describing the necessary experimental
details, we (i) firstly discuss the thickness-dependent characteristic
features of the real-space patterns obtained by MFM in zero field and above
the spin-reorientation transition temperature $T$SR; here we clearly identify
the critical thickness of $D_{0}=4.4$ µm for Mn1.4PtSn plates, indicating the
transition from the FMD to the NCT state. (ii) Secondly, we then focus on the
field dependence and magnetic texture formation for a plate thickness of $D=2$
µm well below the critical thickness $D_{0}$ at temperatures above $T$SR; of
interest here are field-induced phase transitions between the two states.
(iii) Finally, we briefly substantiate and discuss the impact of temperature,
especially when cooling below $T$SR.

##  II Experimental Details

Single crystals of Mn1.4PtSn were grown by flux method using Sn as flux. For
details please refer to Vir et. al [28]. After selecting crystals with
appropriately oriented surfaces, a series of thin-plate samples with
thicknesses ranging from $400$ nm to $10$ µm suitable for MFM were FIB milled
by cutting a $100\times 100$ µm2 groove into the side of the single crystal
using Xe ions at a current of 4 – 60 nA. All samples hence are supported from
the three (bulk) sides resulting in an extremely high mechanical stability for
MFM inspection, as displayed in Fig. 1(A). Note that all sample thicknesses
were checked by scanning electron microscopy with an error of less than 50 nm.
We also checked that gentle sample surface polishing using Xe or Ga ions at
currents below 10 nA, did not impact the magnetic textures of any sample at
all.

Figure 1:  Thickness dependency of the magnetic pattern in Mn1.4PtSn single
crystal at 300K. (A) Side view of a thin-plate sample. The large surface is
oriented normal to the $c$-axis. The inset schematically shows the geometry of
the MFM measurement. (B) MFM image obtained at the end of a 10 µm thick plate,
where it is connected to the bulk (denoted by a red dotted line). The
thickness dependent transformation of the magnetic pattern is clearly visible.
(C1 to C4) MFM images obtained on plates with different thicknesses together
with the corresponding FFT’s plotted on logarithmic color scale (D1 to D4).
Arrows highlight the peaks due to the stripe pattern. While in (C1) and (C2)
the magnetic pattern is close to a sinusoidal wave in (C3) and (C4) the stripe
pattern is closer to a square-wave with more anharmonicity. In the FFTs (D3 &
D4) stripes parallel to the peaks appear due to the nested domains
(highlighted by ellipses). (E) Domain width $W$ plotted as a function of plate
thickness $D$. The first branching is identified at $D_{0}=4.4$ µm. Individual
branches can be fitted by a power law $W(D)=a\cdot D^{b}$.

MFM measurements were performed with two different instruments: Room-
temperature inspection without applying any external magnetic fields was
carried out on the Park Systems NX10 [32] using standard PPP-MFMR probes from
Nanosensors [33] at lift heights between 100 and 150 nm. Low-temperature
measurements with external fields applied, were run in our Omicron cryogenic
ultra-high vacuum STM/AFM instrument [34] operated with the R9 electronics
from RHK [35]. Here we employed PPP-QMFMR probes from Nanosensors driven at
mechanical oscillation amplitudes of $\approx 20$ nm at lift heights between
400 and 800 nm.

##  III Results

###  III.1 Thickness dependence

The domain structure strongly depends on the sample thickness. In Fig. 1(B) we
show the magnetic pattern in zero field at room temperature when inspecting
the border between the 10 µm-thick Mn1.4PtSn plate and the bulk. While the
bulk part exhibits the typical anisotropic fractal domain pattern, the
magnetic pattern seen in the plate area is much simpler showing one row of
nested domains, only. Note, however, that magnetic domain walls in both areas
are aligned along the same preferred crystallographic directions, i.e. along
[100] and [010].

A series of MFM images obtained on thin-plates with different thicknesses is
analyzed in Fig. 1(C). Plates with thicknesses between 10 µm and the treshhold
value $D_{0}=4.4$ µm clearly exhibit stripy patterns with one row of nested
domains [see Figs. 1(C3,C4)]. The corresponding 2-dimensional fast Fourier
transforms (FFTs) (D3), (D4) display two distinct features, (i) a set of
blurred streaks (highlighted by the white ellipse) originating from the nested
domains, and (ii) a series of linearly aligned peaks within each streak
(highlighted by arrows) that correspond to the higher Fourier components of
the non-sinusoidal stripe profile. This behavior changes drastically for
thicknesses below $D_{0}$, where the stripe profiles now become more
sinusoidal showing no nested magnetic domains in real-space images. The FFTs
in turn, show only a one dimensional pattern with less or no higher order
peaks. Thus, $D_{0}$ clearly marks the critical thickness of the first
branching, at which the magnetic pattern fundamentally changes.

From the MFM images we extract line profiles and measure the domain width $W$
or the helix period $\lambda=2W$, respectively. For patterns with nested
domains, we measure the width of both the base domain $W_{b}$ and of the
smallest closure domains $W_{s}$. All measurements are summarized in Fig.
1(E).

The dependence of magnetic domain structure on sample geometry in the case of
thin infinitely extended plates has been described in detail for uniaxial
ferromagnetic systems[31]. In general a power law dependence of the domain
width $W$ on plate thickness $D$ is expected in the form $W(D)=a\cdot D^{b}$,
where constants a and b are tied to the exchange constant $A$, the uniaxial
anisotropy constant $K_{u}$ as well as the saturation magnetization $M_{s}$.
For our experiment, the aspect ratio varies as a function of thickness, and
therefore, we account for varying demagnetization factor $N$ resulting in a
slight thickness dependency of factors a in our analysis. A more detailed
summary can be found in the supplemental material.

In the thickness regime $D<D_{0}$, the domain width $W$ can be well fitted as
a function of plate thickness $D$ by $W=a\cdot D^{1/2}$, as expected for
stripe domains [36, 37, 38] [orange line in Fig. 1(E)]. Yet, leaving the
exponent $\it{b}$ as a free fitting parameter $W\propto D^{b}$, we find
$b=0.65\pm 0.03$, as shown in Fig. 1(E) by the blue solid line. Since in this
regime the observed texture is quite sinusoidal, the stray field is weaker
than for uniform domains of the same width. In contrast to our observations,
for only dipolar interactions this would result in a larger $W$ for constant
$D$. The small deviations from the Kittel-law therefore may be attributed to
the DMI that favors perpendicular spin arrangements and is responsible for the
sinusoidal profile.

For thicknesses $D>D_{0}$, free-exponent fitting for the base domain width
$W_{b}$ results in $b=0.66\pm 0.08$ [green solid line in Fig. 1(E)] in
excellent agreement with the theoretical $b=2/3$-behavior [31]. Meanwhile, the
smallest surface domain width $W_{s}$ remains nearly constant as expected [31,
39]. Thus, we conclude, that for thicknesses above $D_{0}$, Mn1.4PtSn is well
described as a uniaxial ferromagnet with pronounced anisotropy within the
$ab$-plane. Below $D_{0}$, DMI impacts dominantly such that spin helices
instead of stripe domains nucleate.

From $D_{0}=4.4$ µm, the saturation magnetization $M_{s}\approx 4\times
10^{5}$A/m at room temperature[28], and the ratio $K_{u}/K_{d}=1.71$
determined by the fit in the regime $D>D_{0}$ we find $K_{u}\approx
1.71\cdot\mu_{0}\cdot M_{s}^{2}/2\approx 1.71\times 10^{5}J/m^{3}$, which is
small compared to common permanent magnets like Nd2Fe14B or SmCo5 with
$K_{u}=5\times 10^{6}J/m^{3}$ and $K_{u}=1.7\times 10^{7}J/m^{3}$ respectively
[40] and on the same order of magnitude as the dipolar energy.

Thus, we may argue here, that magneto-crystalline and shape anisotropy compete
and provide the effective, thickness-tunable $c$-axis anisotropy. While
Mn1.4PtSn constitutes an easy-axis magnet along the $c$-axis, the shape
anisotropy of a thin plate is thickness dependent and makes the surface normal
of the extended surface to become the magnetic hard axis. For our plate
samples investigated here, the $c$-axis is at the same time the surface
normal, and hence the two anisotropies counteract. For cubic helimagnets with
uniaxial anisotropy, it was predicted that stabilizing non-collinear spin
textures might be possible for selected values of uniaxial anisotropies only
[41, 42]. We thus may speculate that similarly the lowering of the effective
anisotropy is the reason for the thickness-dependent fundamental change of the
reported magnetic textures in Mn1.4PtSn. Therefore, instead of emphasizing on
the relevance of an absolute thickness $D_{0}$, we may rather consider the
critical thickness-to-width aspect ratio $\alpha_{0}$ for our quadratically-
shaped plates as $\alpha_{0}=D_{0}/100\mu\textrm{m}=0.044$. Nevertheless, we
have to add that the special sample shape, where the plate is still connected
to the bulk, may also have an impact on this value.

Figure 2:  Evolution of the magnetic pattern with field applied along the
$c$-axis for a $D=2$ µm thick plate at $T=200$ K. (A1 & A2) Starting from zero
field, a spin helix with field magnitude dependent period persists up to the
critical field strength $B_{c1}=550$ mT. (A3) at $B_{c1}$ antiskyrmions start
to pinch off from helical stripes eventually forming a hexagonal lattice (A4).
(A5) Above $B_{c2}=650$ mT the field polarized state is reached. (A6 to A8)
For decreasing field strength the field polarized state persists until
$B_{c2}=600$ mT, where eventually single antiskyrmions nucleate as seeds for
the helical stripe phase. (B) The apparent helix period as function of
magnetic field strength measured for increasing / decreasing field in blue /
orange color, and for the upper / lower domain with filled / empty symbols,
respectively. (C) The apparent period $\lambda$ can be interpreted as rotation
of the wave-vector $\bm{q}$ into the field with a linear dependency
$\phi(|B|)$.

###  III.2 Field dependence

Next, we focus on the field dependence and evolution of magnetic patterns for
plates with thickness $D<D_{0}$ at $T>T_{SR}$, where ASkLs have been reported.
The field dependent behavior in bulk samples has been discussed in a recent
publication [27]. A series of images measured for $D=2$ µm at $T=200$ K is
illustrated in Fig. 2(A). The complete dataset and datasets obtained at
various other temperatures above $T_{SR}$ are available in the supplemental
material.

Starting from zero external field as shown in Fig. 2(A1), we find a spin helix
up to the critical field strength $B_{c1}=550$ mT. With increasing field
strength a systematic increase of the helix period $\lambda$ is observed [see
Figs. 2(A2&A3)], closely resembling recent findings in Cu2OSeO3 in the helical
phase [43]. Above $B_{c1}$, helical stripes break up into antiskyrmions [see
Fig. 2(A3)], that start to assemble in a hexagonal ASkL with a periodicity of
approximately twice the helix period in zero field, as shown in Fig. 2(A4).
When further increasing the magnetic field, individual A-Skys start to unwind
until the transition to the field-polarized state is reached at
$B_{c2,up}=650$ mT [see Fig. 2(A5)]. In contrast for decreasing field
strengths, the field-polarized state survives until $B_{c2,down}=600$ mT,
where eventually single A-Skys nucleate as seeds for the helical stripe phase
[compare Fig. 2(A6)]. Note, that no ASkL lattice is formed when lowering the
external field from saturation, clearly indicating the hysteretic behavior at
this phase transition. For $B<B_{c2}$, the helical phase is well stable down
to zero magnetic field [see Figs. 2(A7&A8)]. Also note, that reversing the
magnetic field excellently reproduces the above behavior, i.e. with
$B_{c2,up}=-650$ mT and $B_{c2,down}=-600$ mT.

In Fig. 2(B) we analyze the field dependence of the helix period $\lambda$.
Apparently, the helix period changes as a function of external magnetic field
for all domains. Moreover within the helical phase and as seen for instance in
Fig. 2(A1), we find magnetic domains within one and the same image that have
different $\lambda$. In principle, there are two possible explanations for
such a behavior:

Firstly, when assuming that the helix wave-vector $\bm{q}$ is always aligned
in-plane, i.e. always lies in the image plane, then the helix transforms into
an anharmonic helicoid and the magnitude of the period $\lambda$ must
effectively change in magnetic field[41]. Yet, this explanation conflicts with
our observations, as firstly domains with different period are clearly seen,
and secondly the expected field dependency with an approximately constant
period up to a critical field, where the period starts to diverge, does not
fit to our data.

Secondly, a magnitude $\lambda$ that remains constant directly implies tilting
of $\bm{q}$ with respect to the sample surface normal $\bm{n}$, i.e. the
external magnetic field direction. What MFM then measures at the sample
surface, is always the local projection onto the image plane, as has similarly
been reported for the helical state in Cu2OSeO3 [43]. As a result, only the
in-plane component of $\bm{q}$ causes the contrast in MFM. Thus, an apparently
longer period $\lambda=2\pi/[q\cdot\sin(\phi)]=\lambda_{0}/\sin(\phi)$ with
$\phi=\sphericalangle(\bm{q},\bm{n})$ the inclination angle and
$\lambda_{0}=2\pi/q$ the zero field period is measured. Vice versa, from the
values of the apparent helix period $\lambda(B)$, we can compute
$\phi(B)=\arcsin\left(\frac{\lambda_{0}}{\lambda(B)}\right)$. By doing so, we
find an approximately linear dependence $\phi(B)$ for the lower domain [see
Fig. 2(C)]. Extrapolating this linear fit to $\phi(B_{s})=0$ (i.e. the fully
aligned case $\bm{q}||\bm{B}$) yields the critical field $B_{s}\approx 700$
mT, a value that is slightly larger than the onset of the field-polarized
state deduced experimentally here. In contrast to Cu2OSeO3, the tilted helix
becomes unstable already at an angle of $\approx 15$°, long before
$\bm{q}||\bm{B}$.

In small external fields, the upper domain deviates slightly from the linear
behavior, which hints towards an incomplete $\bm{q}$-vector rotation, due to
the very slow dynamics connected to the size of the domain and the large
number of topological defects at the domain boundary. Similar long relaxation
times were also reported for FeGe and MnSi [44, 45]. In total, the field
dependence of the apparent wavelength can be well explained by rotation of the
helix wave-vector $\bm{q}$ into the external field with linear dependence
$\varphi(B)$.

###  III.3 Temperature dependence

Figure 3:  Evolution of the magnetic pattern with field applied along the
$c$-axis for a $D=2$ µm thick plate at $T=80$ K. (A1 - A3) Comparing helix
period and ordering after zero field cooling and after successive field sweeps
up to saturation and back to zero field. The dashed line in (A1) marks the
edge of the plate. (B1 - B3) In the transition to the field-polarized state
only a few isolated bubbles can be spotted as remnants of stripe domains
highlighted by dashed circles.

Finally, we extend our analysis to compare the magnetic pattern evolution in
thin Mn1.4PtSn plates both above and below the spin-reorientation transition
temperature $T$SR at 170 K. In bulk samples, for $T<T_{\text{SR}}$, the
quadratic $ab$-plane anisotropy is lost, resulting in a much less rectangular
domain pattern arrangement[27].

When inspecting the plate with $D=2$ µm, the effects on the magnetic pattern
can be described as follows: Upon zero field cooling to $T=80$ K, the spin
helices evolving above $T$SR, remain. Only when magnetically saturating the
sample and ramping the field back to zero, can we notice some subtle changes
in the magnetic pattern. In more detail, the period slightly increases and
single phase fronts appear less straight, i.e. the helix wave-vector locally
rotates slightly within the $ab$-plane [see Fig. S3(A1 to A3) and Fig. S1 in
the Supplemental Material]. As for bulk samples, the quadratic $ab$-plane
anisotropy characteristic above $T$SR vanishes. Even when ramping the field up
into the field-polarized state showed no evidence of the occurrence of an
ASkL, but only of single antiskyrmions or bubble domains [see Fig.3(B1 to
B3)], in agreement with recent reports on Hall-effect measurements in bulk
samples showing distinct behaviour above and below $T$SR [29].

##  IV Discussion and conclusions

In summary, we performed a thickness-dependent study of the magnetic patterns
observed in Mn1.4PtSn, both under variable magnetic fields and over a wide
temperature range. We found for our quadratic $100\times 100$ µm2-wide plates
the critical thickness $D_{0}=4.4\pm 0.4$ µm corresponding to a critical
thickness-to-width aspect ratio $\alpha_{0}=0.044\pm 0.004$ above which the
material can be well described as a uniaxial ferromagnet with a domain pattern
fully built-up from stripe domains and nested closure domains.

Below $D_{0}$, a spin helix in low fields and an ASkL at fields close to
saturation field were found, in agreement with literature. The phase diagram
shows a strong hysteretic dependence, which manifests in the absence of the
ASkL when the field is decreased from the saturated state. This might be
connected to the observed A-Sky formation by pinching them off from helical
stripes. Note that the reversed process had formerly been identified to unwind
the SkL state towards the topological trivial helical state in Fe0.5Co0.5Si
[11]. The nucleation of further A-Skys in the field-polarized state happens to
be less favorable than the area growth of already nucleated A-Skys into
helical stripes, and hence, no ASkL but a stripe pattern forms. Thus, for
selected magnetic field ranges, stable antiskyrmions may be easily nucleated
by means of an external stimulus without formation of an ASkL at the same
field and temperature. This is quite unusual for single crystalline systems,
but offers the possibility to study single antiskyrmions at will in Mn1.4PtSn.

The apparent period of the spin helix shows a field dependence that can be
well described by the rotation of the spin helix into the field direction,
with a linear dependency of the inclination angle on the applied field. For
temperatures below the spin-reorientation transition temperature the
anisotropy within the $ab$-plane is lost, and the helix wave-vector may
locally rotate within the $ab$-plane.

To conclude, in thin-plate samples of Mn1.4PtSn both the DMI as well as
dipolar interactions are equally important for the stability of antiskyrmions.
By varying the sample thickness and hence the effective $c$-axis anisotropy,
the helical pitch and the ASkL size can be tuned to very large values as
compared to other skyrmion hosting materials. Together with the likely ease to
nucleate single antiskyrmions at will and at any predefined position by means
of an external stimulus without formation of an ASkL, an even wider set of
experimental techniques may be available such as optical detection with
highest temporal resolution for studies of the antiskyrmion formation and
dynamics.

## Acknowledgments

This project was funded by the German Research Foundation (DFG) under Grants
No. EN 434/38-1 and MI 2004/3-1 as well as EN 434/40-1 as part of the Priority
Program SPP 2137 “Skyrmionics”, via the project C05 of the Collaborative
Research Center SFB 1143 (project-id 247310070) at the TU Dresden and the
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter – ct.qmat (EXC 2147, project-id 390858490). B.E.Z.C. acknowledges
support from the International Max Planck Research School for Chemistry and
Physics of Quantum Materials (IMPRS-CPQM).

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The growth and development of living organisms
from the thermodynamic point of view

Alexei A. Zotin
Koltzov’ Institute of Developmental Biology RAS

Vavilov Str. 26, Moscow, Russia 119334

and

Vladimir N. Pokrovskii
Moscow State University of Economics, Statistics and Informatics,

Moscow, RUSSIA 119501

###### Abstract

The living organism is considered as an open system, whereas Prigogine’s
approach to the thermodynamics of such systems is used. The approach allows
one to formulate the law of individual growth and development (ontogenesis) of
the living organism, whereas it has taken into account that the development
and functioning of the system are occurring under the special internal
program. The thermodynamic equation of growth is followed a method of
estimation of the specific entropy of organism. The theory is compared with
experimental data, whereas estimates of the specific entropy of some species
were done; it shows a reduction of specific entropy in the evolutionary row:
yeast - insects - reptiles - birds.

Keywords: Entropy of living organism, Energy exchange during growth,
Thermodynamic equation of growth

October 2018

##  1 Introduction

Many prominent investigators [1 - 4] approached the formulation of
phenomenological (thermodynamic) theory of the development, growth and aging
of a living organism (theory of ontogenesis). The theories were based on the
analysis of quantities characterizing energy exchange between an organism and
the surrounding. Measuring simultaneously both the incoming flux of energy and
intensity of heat production, the investigators found [5, 6] that, under the
growth, the former can surpass the later. The difference between the incoming
flux of energy and intensity of heat production, dubbed as $\psi_{u}$-function
[5, 6], has been qualified as a part of energy that is needed for the
construction and growth of the organism [7]. This idea was central in our
investigation, and we are going now to formulate the exact relation between
$\psi_{u}$-function and the rate of mass growth of the living organism.

From the thermodynamic point of view, the living organism ought to be
considered as an open thermodynamic system that exists in a non-equilibrium
state [4, 8 - 10], and the desire to understand the general principles that
govern the living organisms was one of the important motivations to
investigate open systems. The living organism does not contain anything, apart
of chemical compounds, and all processes that are occurring in the system are
connected, in the large degree, with chemical transformations, so that there
were some attempts to present simple models imitating the behaviour of the
living organism. The brusselator, considered by Prigogine with collaborators
[11], can be considered as one of the elementary models that reflects the
essential features of the biological organism: an exchange of substances with
the environment and existence of stable spatial internal structure within a
limited volume. However, the living organism has extra peculiarities of the
behaviour: unlike the brusselator, all processes are running under internal
control, trying to keep the internal state of organism unchangeable, which is
known as a phenomenon of homeostasis [12]. The control mechanism changes the
speed of running of processes depending on the signals receiving from the
outside. At the rise of outer temperature, for example, the control mechanism
commands to increase perspiration, for the temperature of the organism not to
be increased. This special congenital program also determines the development
and growth of the organism.

Do all of this require some modification of thermodynamic laws at application
to living systems in comparison with chemical systems of the brusselator type?
In this paper, first of all, we are going to remind main results of
Prigogene’s formulation of thermodynamics of open systems (Section 2). In
Section 3 we apply Prigogene’s results to show how the control apparatus can
be introduced into the thermodynamic approach (Section 3.1); this gives a clue
to the problem of the description of growth and development of the living
organism. Then, we are formulating (Section 3.3) and testing (Section 3.4) the
thermodynamic equation of growth of the living organism. The Discussion and
Conclusions section contains a discussion of the resolution of the problem
presented herein.

##  2 The Change of Entropy in Prigogine’s Formulation

The essential contribution to the thermodynamics of open systems was brought
by Prigogine, when he and his collabourators investigated systems of
chemically reacting substances. Stationary states of such systems exist due to
exchange both particles and energy with the environment. In section 8 of the
third chapter of his book [9], Prigogine has specified three contributions to
the variation of entropy of an open system. The simple analysis [13, 14],
reproduced in Appendix, demonstrates that the change of entropy $S$ of the
system at the given temperature $T$ can be calculated according to the formula

$T\,dS=\Delta
Q-\sum_{j}\,\Xi_{j}\,\Delta\xi_{j}+\sum_{\alpha}\,\eta_{\alpha}\,\Delta
N_{\alpha}.$ (1)

The first term on the right-hand side of the equation presents a stream of
thermal energy into the system; it can be positive (the stream into the
system) or negative (the stream out of the system). The last term depicts a
stream of energy coming with the stream of particles of substances $\Delta
N_{\alpha}$ that can be positive (the stream into the system) or negative (the
stream out of the system), $\eta_{\alpha}$ is connected with chemical
potential of substance labeled $\alpha$. The middle term on the right-hand
side of equation (1) presents a change of the stored energy inside the system
due to the existing of internal variables $\xi_{i}$, so called thermodynamic
forces $\Xi_{j}$ are considered positive, so that this term gives a positive
contribution into entropy at relaxation of internal variables. In the case of
chemically reacting substances, which was investigated by Prigogine [9], the
internal variables $\xi_{i}$ appear to be measures of incompleteness of
chemical reactions, that are the measures of how much the considered system
with chemical reactions is out of equilibrium; the thermodynamic forces
$\Xi_{j}$ are affinities of corresponding reactions. The theory was
generalized [13, 14] to consider any deviation from the equilibrium state as
an internal variable, so that we can consider the set of internal variables
$\xi_{j}$ in equation (1) to be comprised of the quantities determing not only
degrees of completeness of all chemical reactions occurring in the system, but
also a structure of the system, gradients of temperature, difference of
concentrations of substances and so on. Note that equation (1) presents the
balance of energy and is not restricted to any approximations.

The sign of the middle term on the right-hand side of equation (1) is taken in
such a way to ensure the positive production of entropy at the relaxation of
internal variables $\xi_{j}$, but two processes are running simultaneously.
Under some external influences (the fluxes of heat and substances) the system
is passing to a new state, while internal variables could emerge and entropy
of the system could change. Due to internal processes that cannot be
controlled from the outside, the internal variables, as measures of non-
equilibrium of the system, tend to disappear. For small deviations of internal
variables $\xi_{i}$ from their equilibrium values $\xi_{i}^{(0)}$, the local
law of disappearing can be written as relaxation equation for each internal
variable

$\frac{d\xi_{i}}{dt}=-\frac{1}{\tau_{i}}\,\left(\xi_{i}-\xi_{i}^{(0)}\right),\quad
i=1,\,2,\ldots,$ (2)

where $\tau_{i}=\tau_{i}(T,x_{1},x_{2},\ldots,x_{n})$ is a relaxation time of
a corresponding internal variable that, being left on its own, permanently
decreases and, eventually, comes close to equilibrium value $\xi_{i}^{0}$,
which is conveniently considered to be equal to zero. In this situation, due
to the recorded equation (2), the middle term on the right-hand side of
equation (1), is positive and describes the dissipation of the stored energy.
The dynamics of internal variables for arbitrary deviations from equilibrium
was formulated in the paper [14].

Note that, in line with the notion of entropy of the open system $S$, one can
introduce the notion of complexity; the latter has the same sense as notion of
bound information [15] in application to complex systems. The complexity can
be considered as a characteristic of the set of internal variables $\xi_{i}$:
the internal variables themselves sometimes are called variables of complexity
or variables of order. Though at the moment it is impossible to define how
complexity is related to the number and connectedness of internal variables,
one can assert that the greater the number of internal variables, their values
and mutual connections, the greater the complexity. In virtue of equation (1),
the change of complexity can be connected with the change of entropy of the
system: increase or decrease in number and values of internal variables
provokes decrease or increase in entropy.

##  3 Organism as an Open Thermodynamic System

We shall discuss the processes of growth and development of the living
organism, considering it as an open thermodynamic system of undefined volume,
but with the certain mass $M$ and temperature $T$. To describe a huge number
of internal processes within the system, one needs a set of internal variables
$\xi_{j}$, that, in case of the living organism, are emerged under the
influence of the streams of substances and heat. On the basis of fundamental
expression for variation of entropy of an open system (1), the equation for
the growth rate of specific entropy $S/M$ of the organism can be written as

$\frac{M}{S}\,\frac{{\rm d}}{{\rm
d}t}\left(\frac{S}{M}\right)+\frac{1}{M}\,\frac{{\rm d}M}{{\rm
d}t}=\frac{1}{TS}\left(\frac{\Delta Q}{\Delta t}-\sum_{i}\,\Xi_{i}\,\frac{{\rm
d}\xi_{i}}{{\rm d}t}+\sum_{\alpha}\,\eta_{\alpha}\,\frac{\Delta
N_{\alpha}}{\Delta t}\right).$ (3)

It is marked with the symbol $\Delta$ that the corresponding quantities are
objects of direct empirical observation.

The law of conservation of mass determines the equation of growth

$\frac{{\rm d}M}{{\rm d}t}=\sum_{\alpha}\,m_{\alpha}\,\frac{\Delta
N_{\alpha}}{\Delta t},$ (4)

where $m_{\alpha}$ is mass of a particle of substance labeled $\alpha$.
Unfortunately, this form of the growth equation is useless practically, so as
there are a lot of the processes occurring in the living system, and it is
extremely difficult to measure the substances coming in and out the organism
[4].

The balance equations (3) and (4) give a basis for thermodynamic
interpretation of the phenomena of existence, growth and ageing of a
biological organism, but for this purpose, it is necessary to establish, first
of all, correspondence of all symbols in these parities to the quantities
concerning the living organism.

###  3.1 Three groups of internal variables

Let’s remind, that the internal variables that are present in the middle term
of the right part of the equation (3) determine the deviation of the system
from the state of equilibrium and, in this way, apart of the incompleteness of
internal chemical reactions, describe the architectural and functional
structure of the system. Considering the organism as an open system, we can
note that there is permanent changing of internal variables, which emerge
under fluxes of heat and substances and tend to disappear. The rate of
disappearing is different for different internal variables; time of relaxation
of some of them are very small as compared with the lifespan of organism. The
other internal variables exist during the total lifetime and even more. The
difference of internal variables according to their rates of approach the
equilibrium state determines their functional distinctions in the processes of
the growth and development of the organism.111Consideration of an hierarchy of
relaxation processes is useful in many applications of non-equilibrium
thermodynamics. For example, Gujrati [16] demonstrated recently that the
division of internal variables according to their relaxation times is very
helpful for description of the phenomena of solidification of glasses. For the
further discussion, we are dividing the set of internal variables of the
system into three groups, according to values of relaxation times; further on,
the different functional roles of separate groups in the processes of
development and growth are described.

First of all, it is possible to separate a group of internal variables, which
originates from the complexity of the first cell that gives the beginning of
the organism. The material carriers of complexity are the molecules of a
deoxyribonucleic acid packed in the nuclei of the cells. The copies of
internal variables (complexity) of the initial set are present in each cell
and remain constant (at some idealization) up to the death of an organism. The
number of copies of these variables increases at the proliferation of cells,
and during the realization of the program of development. These variables
comprise the foundation of a control mechanism that supervises all processes
in the organism during its existence and growth. The organism is developing
under the installed plan received by right of succession, whereas some
internal variables are suppressed or expressed when the cells are multiplying.
A role of internal variables of the first group is the conservation of
hereditary instructions and the control over the construction of an organism.

The second group of internal variables describes the emerged structure of the
organism in processes of morphogenesis and differentiation, when the
interoperability of the parts of an organism is materializing. For example,
conformations of the biomolecules having albuminous in the basis, change in
such a manner that speed of reactions in which they participate, also sharply
changes. The greater role is also playing a spatial arrangement of fibers and,
in particular, enzymes [17]; it provides such processes that cannot occur in a
random mixture of chemical reagents. The number and values of such internal
variables increase at growth and development of the organism. The set of
variables of the second groups determines some metastable structure (organs
and tissues) of the living organism and have the times of relaxation
comparable with the lifespan of the organism, so that they do not contribute
into the current dissipation of energy (production of entropy).

The set of internal variables of the third group emerges in the course of the
above-described processes of multiplication of original complexity and
construction of the structure of the organism. The processes are based on
chemical transformations of substances, occur in non-equilibrium situations
and are accompanied by excitation of internal variables and their fast
relaxation. The set of internal variables of this group includes the variables
locating gradients of temperature, differences in concentration of substances
and degrees of completeness of all chemical reactions running in the organism.
Internal variables of the third group accompany all existing processes
anywhere they were. These internal variables have very small times of
relaxation (in comparison with existence of the organism) and determine the
basic contribution to the internal production of entropy. The dissipated
energy is coming out in the form of heat.

Note that now we can record an expression for entropy of the organism as a
function of the internal variables, whereas the expansion can be done over
internal variables of the third group $\mbox{\boldmath$\xi$}^{(3)}$

$S(\mbox{\boldmath$\xi$}^{(1)},\mbox{\boldmath$\xi$}^{(2)},\mbox{\boldmath$\xi$}^{(3)})=S_{0}(\mbox{\boldmath$\xi$}^{(1)},\mbox{\boldmath$\xi$}^{(2)})-\,\frac{1}{2T}S_{ij}(\mbox{\boldmath$\xi$}^{(2)})\,\xi^{(3)}_{i}\xi^{(3)}_{j}.$

The quantity $S_{0}$ is entropy of a dead body (a corpse), the quantity is a
decreasing function of variables of the first group
$\mbox{\boldmath$\xi$}^{(1)}$ (carriers of instructions for construction and
functioning of an organism) and the second group $\mbox{\boldmath$\xi$}^{(2)}$
(the description of metastable structures of the organism). The last term on
the right hand side of the equation presents all processes running in the live
organism; the quantities $S_{ij}$ are positive and essentially depend on
internal variables of the second group, which define metastable structure (a
set of structures) of the system and cannot be considered as small quantities.

The difference in the relaxation times of internal variables allows us to
distinguish different functional roles of internal variables in the processes
of development and growth. The distribution of the all internal variables over
three groups can be helpful for designing of the functional scheme of the
living organism.

###  3.2 Exchange of energy between the organism and environment

There are methods of assessments of incoming and outgoing fluxes of energy for
the organism of an animal [4]; these fluxes are presented in the right part of
the equation (3) by the first and the last terms. The greatest amount of
energy is delivered into organisms with food in the form of energy of chemical
links of organic molecules [4, 18]. The speed of synthesis of adenosine
triphosphate (ATP) gives us an estimate of the coming-with-food energy (in a
unit of time). The synthesis can be realized in the two basic processes: the
oxidizing phosphorylation and glycolysis. One can assume that the energy
received in the process of glycolysis, can be neglected in comparison with the
process of oxidizing phosphorylation [4], and one can judge about the speed of
synthesis ATP indirectly, namely on the rate of oxygen consumption or the rate
of allocation of carbonic gas. It is shown, that quantity of the consumed
oxygen is proportional to the energy coming into the organism, whereas the
factor of proportionality (so-called oxi-caloric coefficient) is approximately
equal to $4.821\;cal/ml\;{\sf O}_{2}$ [19]. The estimating of incoming energy
by the speed of consumption of oxygen has received the name the method of
indirect calorimetry.

The energy of molecules of ATP is used basically for the synthesis of
molecules, which are included in the organism, also as for other processes
requiring expenses of energy (muscle contraction, migration of cells,
conduction of nervous impulses, etc.). Regarding the above-described
classification of internal variables, the result of the release of energy of
molecules of ATP is the emergence of internal variables of the second group,
which is accompanied by excitation of internal variables of the third group.

The flux of heat energy from the living system $\Delta Q$ is measured by the
method of direct calorimetry, that is by direct measurement of the speed of
heat production of the organism by means of a calorimeter. The amount of
thermal energy coincides with the dissipation of energy that appears at the
relaxation of internal variables of the third group with small times of
relaxation. The amount of the heat flux is evidence of the presence of quickly
relaxating internal variables.

In steady state, when there are no changes of thermodynamic functions,
incoming and outgoing flows of energy, according to equation (1), are equal.
When an organism grows, the incoming stream of energy provokes the emergence
of internal variables of the second and third group. The relaxation of
variables of the third group defines dissipation of energy that in thermal
form is deleted completely from the organism. Remind that the internal
variables of the first and second groups do not contribute to the dissipation
due to their huge relaxation times. Thus, the difference between the incoming
and outgoing flows, if exists, gives rise to the emergence of structural
variables of the second group, providing the construction of organs and
tissues of the organism.

Entropy $S$ of the whole organism, due to the mass growth, increases;
simultaneously increases in the number of internal variables and their mutual
connections that is, one can say, increases in structural complexity of the
organism. The better, than the total entropy $S$, characteristic of the living
organism is the specific entropy $S/M$: under the balance of incoming and
outgoing energy, specific entropy $S/M$, as it follows from equation (3), can
only decrease. In the situations of growth, the complexity of organism
increases, and, so as the greater the complexity, the less entropy, one can
assume that the specific entropy does not increase, and it gives

$\frac{M}{S}\,\frac{{\rm d}}{{\rm
d}t}\left(\frac{S}{M}\right)=-\sigma,\quad\sigma\geq 0.$ (5)

At the moment we cannot say that the quantity $\sigma$ is constant, but later
(in Section 3.4), its value will be estimated and appears to be about zero.

The permanent exchange of substances and energy between the organism and
environment, which is a necessity for the creation of structures and
maintenance of functioning, is controlled by the program (the internal
instructions) stored with internal variables of the first group.

###  3.3 Equations of growth

From the previous reasoning, it follows that the growth rate of mass of the
organism is essentially determined by a difference between quantities of
incoming energy and outgoing heat. Equation (3) shows the linear dependence of
the specific growth rate of mass on the difference, which had attracted
special attention and attained the name of $\Psi_{u}$-function [5]. The
specific psi-function is defined as

$\psi_{u}=\frac{1}{M}\left(\frac{\Delta Q}{\Delta
t}-\sum_{i}\,\Xi_{i}\,\frac{{\rm d}\xi_{i}}{{\rm
d}t}+\sum_{\alpha}\,\eta_{\alpha}\,\frac{\Delta N_{\alpha}}{\Delta t}\right).$
(6)

Now one can reformulate relation (3), using definitions (5) and (6), so that
the equation of growth can be recorded in the form

$\frac{1}{M}\,\frac{{\rm d}M}{{\rm d}t}=\sigma+\kappa\,\psi_{u},$ (7)

where $\kappa=M/TS$. According to the above equation, the origin of growth is
the difference between quantities of incoming and outgoing flows of energy.
The growth occurs until the organism will not reach the final steady state
defined both by the genetic program, and conditions of the inhabitancy. The
majority of poikilothermal animals grow during all life. The homoiothermal
animals (mammals and birds) and some of the poikilothermal ones reach the
stable mass [20].

In the stationary case the organism should be considered as a system, in which
all processes appear balanced:

$\frac{\Delta Q}{\Delta t}-\sum_{i}\,\Xi_{i}\,\frac{{\rm d}\xi_{i}}{{\rm
d}t}+\sum_{\alpha}\,\eta_{\alpha}\,\frac{\Delta N_{\alpha}}{\Delta t}=0.$ (8)

The adult organism, thus, can be considered as the certain constant structure
that is being in the non-equilibrium stable state described constant
thermodynamic functions.

The time dependence of the mass growth for all types of animals can be
approximated by a simple equation [21]:

$\frac{{\rm d}M^{1/u}}{{\rm d}t}=V_{0}c^{t},\quad 0<c<1,$ (9)

where $V_{0}$ is the speed of the growth in initial point in time $t=0$; $c$
is a parameter determining the rate of growth; $u$ is a correction factor that
is taking into account interfaced processes (morphogenesis, differentiation,
etc.).

The special case of equation (9) - the equation of Bertalanffy [22] – can be
recorded in the form

$\frac{1}{M}\,\frac{{\rm d}M}{{\rm d}t}=\frac{u\ln c}{c^{t-t_{0}}-1},$ (10)

where $t_{0}$ is a conditional time, at which assumingly $M=0$, $c$ is a
factor determining the rate of growth, $u$ is a factor connected with the
influence of morphogenetic processes during the growth.

###  3.4 Testing of the thermodynamic equation of growth

To estimate correspondence of equation (7) to reality, we address to results
of direct observation of growth of animals in the situations, when one can
control the fluxes of energy between the organism and environment [4, 5, 23 -
28]. The fluxes of heat from the organism were measured by the method of
direct calorimetry. The fluxes of chemical energy in the organism were
estimated by the method of indirect calorimetry according to the rate of
oxygen consumption. We can draw certain conclusions from the results.

It has appeared, that during an essential period of ontogenesis of organisms,
the values of incoming and outgoing energy fluxes practically coincide,

Figure 1 The dependence of specific growth rate on $\psi_{u}$-function

Species: a \- yeast Saccharomyces cerevisiae (according to Schaarschmidt et
al. [24]; b \- larvae of crickets Acheta domestica; c \- embryos of lizards
Lacerta agilis; d \- chicken embryos Gallus domesticus (according to Kleymenov
[28]). The dependences are approximated by straight lines, according to
equation (7).

Table of Estimates of Specific Entropy

Type (class) | $\kappa,\;g/mW\cdot day$ | $\sigma,\;day^{-1}$ | $r^{2}$ |  $\;S/M$, $J/g\cdot K$
---|---|---|---|---
Saccharomyces cerevisiae (Saccharomycetes)1 | $(4.73\pm 0.43)\cdot 10^{-7}$ | $(5.70\pm 2.98)\cdot 10^{-6}$ | $0.92\pm 0.08$ | $365\cdot 10^{6}$
Acheta domestica  (Insecta), larvae2 | $0.042\pm 0.009$ | $0.011\pm 0.014$ | $0.76\pm 0.16$ | 4400
Lacerta agilis  (Reptilia), embryos2 |  0.062 $\pm 0.016$ | $0.015\pm 0.046$ | $0.87\pm 0.22$ | 3350
Gallus domesticus (Aves), embryos2 | $0.166\pm 0.046$ | $0.123\pm 0.098$ | $0.98\pm 0.07$ | 300

Sources of data: 1 – Schaarschmidt et al. [24]; 2 - Kleymenov [28].

and such a situation has received the name of the current steady state of the
organism [4, 7, 18]. However, there is mass growth in such situations, and,
according to equation (7), the difference between incoming and outgoing flows
of energy must exist. One can assume that the growth occurs so slowly, that
the difference appears to be so small, that, apparently, cannot be detected
with the used methods. Apart of it, when the method of indirect calorimetry is
used for the assessment of incoming stream of energy, the contribution of
processes of glycolysis, which are also processes of synthesis of ATP, are
neglected in comparison with the process of oxidizing phosphorylation.
However, the contribution of the processes of glycolysis can appear
comparatively essential in this phase of growth.

The authentic distinction between data of the indirect (according to the rate
of oxygen consumption) and the direct (according to the rate of heat
production) calorimetry in situations of the constant environment is observed
in cases of the initialization of such processes, as differentiation,
morphogenesis etc [28]. At early stages of the growth in ontogenesis, the
distinction of results of the direct and indirect calorimetry $\psi_{u}$ is
great enough to address to an assessment of the adequacy of equation (7).

Figure 2 The time dependence of $\psi_{u}$

Species: a \- yeast Saccharomyces cerevisiae (according to Schaarschmidt et
al. [24]); b \- larvae of crickets Acheta domestica; c \- embryos of lizards
Lacerta agilis; d \- chicken embryos Gallus domesticus (according to Kleymenov
[28]). The dependences are approximated by curve lines, according to equation
(11).

The specific growth rates, as functions of $\psi_{u}$, according to
experimental data [24, 28] are shown on Fig. 1. The values of coefficients of
correlation and parameters of equation (7) are collected in Table. Though the
variability of data is great, the coefficient of correlation between the
$\psi_{u}$ and specific growth rate differ reliably from zero ($p<0.01$). Note
that the quantity $\sigma$ is not significantly different from zero
considering the error bars. Thus, specific entropy can be considered as a
constant during a significant time interval, and the calculated values of the
coefficient $\kappa$ allow to estimate specific entropy $S/M=1/\kappa\,T$ (see
the Table). It is seen that specific entropy decreases in the evolutionary
raw: yeast $\rightarrow$ insects $\rightarrow$ reptiles $\rightarrow$ birds.
These results show a reduction of entropy or, in other words, an increase of a
degree of complexity of living systems during evolution.

A positive correlation between the specific growth rate and $\psi_{u}$ also
has been noted for representatives of different taxons: bacteria Serratta
marinorubta, S. mercescens [27], yeast Sacharamyces cerevisiae [24], larvae of
mealworm beetle Tenebrio molitor [25], larvae of house crickets Acheta
domestica, embryos of sand lizard Lacerta agilis, chicken embryos Gallus
domesticus [28], juvenile fishes Xiphophorus helleri [23], larvae of axolotl
Ambystoma mexicanum [26], ovogenesis of clawed frogs Xenopus laevis [5]. It is
important to have a look at the time dependence of the quantity $\psi_{u}$ at
the growth of the organisms: it gives an additional way of assessment of
adequacy of the equation (7). Comparing equation (10) with the equation (7),
we receive:

$\psi_{u}=\frac{a}{1-c^{t-t_{0}}}-b,$ (11)

where $a=-u\ln c/\kappa;b=\sigma/\kappa$. On the plots of Fig. 2, the
experimental values of $psi$-function, corresponding to values in Fig.1̇, are
shown with empty circles. The solid curves are calculated, according to
equation (11), at individual values of quantities $a$, $b$ and $c$. This
comparison confirms the existence of the universal law of growth in the form
(10).

##  4 The Discussion and Conclusions

The thermodynamic approach for the open systems, developed by Prigogine [9]
and his followers, gives a general foundation for description of dynamics of
processes running in living systems. The internal variables, the number of
which is huge, describe the deviations of the biological organism, as a
thermodynamic system, from equilibrium and appears to be the most essential
elements of description. However, we cannot even inventory all internal
variables, and are still very distant from the detailed description of the
living organism. We have found that, though the biological organism exists as
integral system and all internal variables are anyhow connected with each
other, it is useful to allocate some groups of the internal variables
according to the relaxation times. The mechanisms of heredity and control,
which is the main thing that distinguish the biological organism from a system
of coupled chemical reactions, is assumed to be ascribed to a special group
(the first group in our classification) of internal variables. The internal
variables with relaxation times comparable with the lifespan of organism (the
second group) describe the metastable structure that is a real implementation
of living organism. The variables of the third group are participating in
thermodynamic processes that are obeying the linear laws of the non-
equilibrium thermodynamics.

The clarification of functional roles of the internal variables allows us to
formulate the thermodynamic equation of growth of organisms that corresponds
not only to the results of experimental observations but also to basic
phenomenological equations of growth. It is remarkable that application of the
derived equation of growth to experimental data allows one to estimate the
value of specific entropy of living organisms.

The specific entropy appears to be the most important thermodynamic
characteristic of the living organism. In some examples, it was shown that
specific entropy remains constant during the growth and development of the
organism. It is possible that this property is valid only for many-cell
bodies, when the growth is connected with the multiplication of the high
complexity of the original cell. So as the specific entropy does not change
substantially during the lifespan, it appears to be an individual
characteristic of the organism. The assessments of specific entropy for larvae
of crickets, embryos of the sand lizard and chicken embryos confirm that there
is decrease of specific entropy and, consequently, an increase of a degree of
complexity of the living systems during progressive macroevolution.

## Appendix: Entropy of the open thermodynamic system

The concept of entropy, as a measure of the degraded energy, has arisen at the
description of the ideal equilibrium situations. Variation of entropy $S$
under reversible process of a heat transaction to system in the amount $\Delta
Q$ is defined as

$\Delta S=\frac{\Delta Q}{T},$ (A.1)

This circumstance allows to interpret entropy of the systems in equilibrium
situations as a measure of thermal, completely degraded energy under some
restrictions by a set of constitutive parameters (volume $V$ in the simplest
case) that are defined the thermodynamic system. At the extension of concept
of entropy for non-equilibrium states of the system, it is natural to demand,
that interpretation of entropy as a measure of thermal, completely degraded
energy is being kept. The ignoring of this requirement has brought to
existence of a set of ’non-equilibrium entropies’ and appears to be a source
of some misinterpretation.

We shall demonstrate the realization of such an approach, following the works
[13, 14], for general case of open system, which is assumed performs no work.
Then, the expression for variation of total internal energy $U$ (the first law
of thermodynamics) is recorded in the form of

$\Delta U=\Delta Q+\sum_{\alpha}\,(\eta_{\alpha}+\mu_{\alpha})\,\Delta
N_{\alpha},$ (A.2)

where $\Delta Q$ is a flux of heat into the system, $\Delta N_{\alpha}$ is a
change of substance of a kind $\alpha$ within the system. The enthalpy of
unity of the substance of kind $\alpha$ is presented as a sum of two
quantities, $\eta_{\alpha}+\mu_{\alpha}$, one of them $\mu_{\alpha}$ is
chemical potential of the substance.

The parity (A.2) is valid both for equilibrium, and for non-equilibrium
situations, however, if, in equilibrium states, internal energy $U$ represents
completely degraded energy, in a non-equilibrium situation only some part of
the total internal energy $U$, which we shall designate as $E$, is in the form
of thermal energy. Other part of energy is connected with the restrictions
caused by constant constitutive parameters (volume $V$ in the simplest case)
and relaxating internal parameters $\xi$. The latter are introduced to
describe deviations of the thermodynamic system from the equilibrium state.
The total internal energy of the system changes due to the change of the
composition of the system222Thanks to W. Muschik, who pointed at the existence
of this contribution. and work on variations of the internal parameters $\xi$,
so that it can be recorded as

$\Delta U=\Delta
E+\sum_{j}\,\Xi_{j}\,\Delta\xi_{j}+\sum_{\alpha}\,\mu_{\alpha}\,\Delta
N_{\alpha}.$ (A.3)

The part of total internal energy is in the thermal form, and a part in the
form that is ready to turn in thermal energy.

There are various ways of definition of entropy for the open thermodynamic
system in view of all of non-equilibrium processes. In the simplest version,
one keeps definition (A.1), which, apparently, is wrong, because since times
of Clausius is known that in non-equilibrium processes for the closed system

$\Delta S\geq\frac{\Delta Q}{T}$ (A.4)

In view of it, the definition of non-equilibrium entropy for the closed system
should contain additional positive terms

$\Delta S=\frac{\Delta Q}{T}+....$

To define entropy as a measure of thermal energy, we shall return to the law
of conservation and transformation of energy for the system, which can be
recorded, according to parities (A.2) and (A.3) in the form of

$\Delta E=\Delta
Q-\sum_{i}\,\Xi_{i}\,\Delta\xi_{i}+\sum_{\alpha}\,\eta_{\alpha}\,\Delta
N_{\alpha}.$ (A.5)

Further, adhering to interpretation of entropy as a measure of energy in the
form of heat, we follow an equation (A.1) and, neglecting the work of the
system, record

$\Delta S=\frac{\Delta E}{T}.$ (A.6)

Here temperature $T$ is also defined as absolute temperature for the thermal
part of the system that has thermal energy $E$. Of course, the real
temperature of the system in a non-equilibrium state could differ from this
temperature $T$, but the difference can be considered as an internal variable
and included in the list of $\xi$(inexplicitly).

In such a way, equation (A.5) and (A.6) determines the differential of entropy

$\displaystyle dS$ $\displaystyle=$ $\displaystyle\frac{1}{T}\,\left(\Delta
Q-\sum_{i}\,\Xi_{i}\,\Delta\xi_{i}+\sum_{\alpha}\,\eta_{\alpha}\,\Delta
N_{\alpha}\right).$ (A.7)

Equation (A.7) is the only definition of entropy of the open system, which is
consistent with the understanding this quantity as a measure of the degraded
energy and empirical evidences.

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