Radiation-entropy generation by leaf and negentropy build-up of plant as dissipative structure

Abstract

Photosynthesis is a simple reaction for any plant, but it has not been achieved artificially due to the difficulty in radiation entropy production by processing PAR (photosynthetically active radiation). In this study, a plant is shown to be a self-organising dissipative structure; since, it’s leaf generates radiation entropy for the existence of plant with possible growth. Entropy-energy ratio (s_E) of PAR released after processing (s_E,rel) much exceeds s_E of light absorbed, s_E,in (< < s_E,rel). Plant builds its negentropy $\left|s_{\text{neg},\text{rad}}\right|$ by the difference, $s_{\text{E,rel}}-s_{\text{E,in}}$ , but only a part of this difference can be the source of free energy as plant-growth. The remaining part of $\left|s_{\text{neg},\text{rad}}\right|$ is dissipated as the excess payment of negentropy debt, towards mandatory 2nd Law compliance. Change from low s_E,in to high s_E,rel by the plant-leaf is amplified by a huge factor, c², while determining $\left|s_{\text{neg},\text{rad}}\right|$; where, c is the speed of light. Therefore, even a small increase in s_E,rel and small reduction in s_E,in can significantly increase $\left|s_{\text{neg},\text{rad}}\right|$. Thermodynamic performance of photosynthesis depends on the processing level of PAR, ${\mathrm{\Pi }}_{\text{E},\text{leaf}}=\left(s_{\text{E},\text{rel}}\left./\phantom{s_{\text{E},\text{rel}}{s}_{\text{E},\text{in}}}\right)s_{\text{E},\text{in}}\right)$, $\left|s_{\text{neg},\text{rad}}\right|$, and is limited by the maximum photosynthetic efficiency in PAR, ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}$ (it increases with ${\mathrm{\Pi }}_{\text{E},\text{leaf}}$).

Graphical Abstract

Introduction to thermodynamics of photosynthesis

Life forms, including plants are systems that are formed and exist far from global thermodynamic equilibrium. In this regime, entropy production is governed by the Law of Maximum Entropy Production (LMEP) [1], also known as the Maximum Entropy Production Principle (MEPP) [2]. The Least Action Principle (selection of available net path of least resistance to entropy production) when applied to dissipative systems reduces to LMEP/MEPP [3].

Background on self-organizing dissipative structure (DS) from 1st law and 2nd law

Irreversible processing of mass-energy in-flows taken from the surroundings by a Dissipative Structure (DS) [4] generates net entropy at rate ($\mathrm{\Delta }{\dot{S}}_{\text{gen},\text{sur}}$) in the surroundings (ref. Table 1 for definitions of scientific terms in this study). These mass-energy interactions maximize the entropy production globally, considering DS and its surroundings together. Self-organizing DS, as schematically illustrated in Fig. 1a, has the following thermodynamic features:

• (i)Mass content m_DS and energy content E_DS. Entropies of mass and energy content of DS are, ${\text{S}}_{\text{DS}}\left(={\text{m}}_{\text{DS}}·{\text{s}}_{\text{DS}}\right)$ and ${\text{S}}_{\text{E},\text{DS}}\left(={\text{E}}_{\text{DS}}·{\text{s}}_{\text{E},\text{DS}}\right)$ ; where, s_DS (entropy per unit mass) is entropy density of m_DS and s_E,DS (entropy per unit energy) is entropy density of E_DS.
• (ii)Mass and energy flow rates taken in by DS from its surroundings are, ${\dot{m}}_{\text{in}}$ and ${\dot{E}}_{\text{in}}$. Entropy in-flow rates associated with ${\dot{m}}_{\text{in}}$ and ${\dot{E}}_{\text{in}}$ are, ${\dot{\text{S}}}_{\text{in}}\left(={\dot{m}}_{\text{in}}·{\text{s}}_{\text{in}}\right)$ and ${\dot{\text{S}}}_{\text{E},\text{in}}\left(={\dot{E}}_{\text{in}}·{\text{s}}_{\text{E},\text{in}}\right)$.
• (iii)Mass and energy flow rates released to the surroundings after processing by DS are ${\dot{m}}_{\text{rel}}$ and ${\dot{E}}_{\text{rel}}$. Entropy release rates associated with ${\dot{m}}_{\text{rel}}$ and ${\dot{E}}_{\text{rel}}$ are, ${\dot{\text{S}}}_{\text{rel}}\left(={\dot{m}}_{\text{rel}}·{\text{s}}_{\text{rel}}\right)$ and ${\dot{\text{S}}}_{\text{E},\text{rel}}\left(={\dot{E}}_{\text{rel}}·{\text{s}}_{\text{E},\text{rel}}\right)$.

Table 1.

Glossary of important thermodynamic terms used in this study

Term	Description
Dissipative structure (DS)	Phrase coined by I. Prigogine [4]: Refers to localised ordering that can exist far from global thermodynamic equilibrium, to increase the global free energy dissipation rate
Entropy	Measure of disorder or randomness
Global equilibrium	Zero unbalanced potentials (or driving forces) within an isolated system
Free energy	Energy in a physical system, which can be harnessed for useful mechanical work
Photosynthetically active radiation (PAR)	Range of solar radiation (0.4 to 0.7 μm) that photosynthetic organisms can use for photosynthesis. Energy of photons at shorter wavelengths (< 0.4 μm) can be damaging to the cells and tissues, leading to injury of the leaf. Photons at longer wavelengths (> 0.7 μm) carry insufficient energy for photosynthesis to occur
Law of maximum entropy production	Isolated System away from equilibrium will select least resistance path or assemblage of paths out of available paths that minimizes the potential or maximizes the entropy at the fastest rate for given constraints [1]

Fig. 1.

Schematic of net entropy production in the surroundings by self-organized plant as DS. a Schematic of self-organized dissipative structure (DS, e.g. plant). b Plant as DS producing net radiation entropy by processing PAR (photosynthetically active radiation)

In-flow rates in DS (e.g. plant), ${\dot{E}}_{\text{in}}$ and ${\dot{m}}_{\text{in}}$, are split in two separate parts, for processing of radiation energy (as in photosynthesis) and matter. By the conservation of mass and energy, DS growth rates are given as;

$${\dot{E}}_{\text{DS}}={\dot{E}}_{\text{in}}-{\dot{E}}_{\text{rel}},\text{and}{\dot{m}}_{\text{DS}}={\dot{m}}_{\text{in}}-{\dot{m}}_{\text{rel}}$$ 1

Absorbed matter/radiation energy, ${\dot{m}}_{\text{DS}}$ > 0 and ${\dot{E}}_{\text{DS}}$  > 0, indicate growth of DS (${\dot{m}}_{\text{DS}}$ < 0, ${\dot{E}}_{\text{DS}}$ < 0, indicate DS-decay). From Eq. (1), the efficiency of using in-flows for DS-growth is given as;

$$\begin{array}{cc}& {\eta }_{\text{grow},\text{E}}=1-\left({\dot{E}}_{\text{rel}}\left./\phantom{{\dot{E}}_{\text{rel}}{\dot{E}}_{\text{in}}}\right){\dot{E}}_{\text{in}}\right)=\left({\dot{E}}_{\text{DS}}\left./\phantom{{\dot{E}}_{\text{DS}}{\dot{E}}_{\text{in}}}\right){\dot{E}}_{\text{in}}\right),\text{and}\hfill \\ \hfill & {\eta }_{\text{grow},\text{m}}=1-\left({\dot{m}}_{\text{rel}}\left./\phantom{{\dot{m}}_{\text{rel}}{\dot{m}}_{\text{in}}}\right){\dot{m}}_{\text{in}}\right)=\left({\dot{m}}_{\text{DS}}\left./\phantom{{\dot{m}}_{\text{DS}}{\dot{m}}_{\text{in}}}\right){\dot{m}}_{\text{in}}\right)\hfill \end{array}$$ 1.1

Thus, ${\mathrm{\eta }}_{\text{grow},\text{E}}$ (= ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ , for plant-leaf) is the fraction of radiation energy input that is used by DS for its growth; where,

$$\begin{array}{cc}& 0<{\mathrm{\eta }}_{\text{grow},\text{E}}<{\left({\mathrm{\eta }}_{\text{grow},\text{E}}\right)}_{\text{max}}<1,\text{and}\hfill \\ \hfill & 0<{\mathrm{\eta }}_{\text{grow},\text{m}}<{\left({\mathrm{\eta }}_{\text{grow},\text{m}}\right)}_{\text{max}}<1,\hfill \end{array}$$ 1.1.1

and for decay: ${\mathrm{\eta }}_{\text{grow},\text{E}}<0$ and ${\mathrm{\eta }}_{\text{grow},\text{m}}<0$.

For 2nd Law compliance in the surroundings, the release flows, ${\dot{E}}_{\text{rel}}$ and ${\dot{m}}_{\text{rel}}$, pay the excess negentropy debt for DS existence, by generating net entropy [5]:

$$\begin{array}{cc}\hfill \mathrm{\Delta }{\dot{S}}_{\text{gen,sur}}=& \left({\dot{S}}_{\text{E,rel}}-{\dot{S}}_{\text{E,in}}\right)+\left({\dot{S}}_{\text{rel}}-{\dot{S}}_{\text{in}}\right)\hfill \\ \hfill =& \left({\dot{E}}_{\text{rel}}·s_{\text{E,rel}}-{\dot{E}}_{\text{in}}·s_{\text{E,in}}\right)\hfill \\ \hfill & +\left({\dot{m}}_{\text{rel}}·s_{\text{rel}}-{\dot{m}}_{\text{in}}·s_{\text{in}}\right)\ge 0.\hfill \\ \hfill \end{array}$$ 2

If $\mathrm{\Delta }{\dot{S}}_{\text{gen},\text{sur}}=0$, then no excess negentropy debt is paid by DS to its surroundings; in this case, ${\dot{S}}_{\text{E},\text{rel}}={\dot{S}}_{\text{E},\text{in}}$, and ${\dot{S}}_{\text{rel}}={\dot{S}}_{\text{in}}$. In Eq. (2), the entropies, S_E, S, are extensive properties, which are written in terms of the respective intensive properties: s_E (= entropy-energy ratio, entropy density of energy), s (= specific entropy, entropy density of mass). For net radiation entropy production in the surroundings by radiation exchange ($\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}$), the applicable entropy density is s_E (used in this study). Inequality in Eq. (2) is separated into two terms for radiation and matter exchanged as;

(using Eq. (1)),

$$\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}={\dot{E}}_{\text{rel}}·s_{\text{E},\text{rel}}-{\dot{E}}_{\text{in}}·s_{\text{E},\text{in}}=\left(s_{\text{E},\text{rel}}-s_{\text{E},\text{in}}\right)·{\dot{E}}_{\text{rel}}-s_{\text{E},\text{in}}·{\dot{E}}_{\text{DS}}\ge 0$$

$$\mathrm{\Delta }{\dot{S}}_{\text{mat},\text{gen}}={\dot{m}}_{\text{rel}}·s_{\text{rel}}-{\dot{m}}_{\text{in}}·s_{\text{in}}\ge 0.$$ 2.1

Dimensionless ratios of exit to inlet entropy densities:

$${\mathrm{\Pi }}_{\text{E},\text{DS}}=\left(s_{\text{E},\text{rel}}\left./\phantom{s_{\text{E},\text{rel}}{s}_{\text{E},\text{in}}}\right)s_{\text{E},\text{in}}\right)>>1,{\mathrm{\Pi }}_{\text{m}}=\left(s_{\text{rel}}\left./\phantom{s_{\text{rel}}{s}_{\text{in}}}\right)s_{\text{in}}\right)>>1;$$ 2.2

are the measures/indicators of processing level of in-flows by self-organizing DS. From Eq. (2.1), growth rate of self-organizing DS [${\dot{E}}_{\text{DS}}>0,{\dot{m}}_{\text{DS}}>0$, ref. Equation (1)] is increased by reducing the release flows for given in-flows as;

$${\dot{E}}_{\text{rel},\text{min}}<{\dot{E}}_{\text{rel}}<{\dot{E}}_{\text{in}},{\dot{m}}_{\text{rel},\text{min}}<{\dot{m}}_{\text{rel}}<{\dot{m}}_{\text{in}}$$ 2.3

Released mass-energy content can be lower than the in-flows due to processing by DS, to enable DS-growth; but the inequalities in Eq. (2.1) must be satisfied regardless of DS-growth. Therefore, release flows must exceed a minimum threshold, ${\dot{E}}_{\text{rel},\text{min}}$, ${\dot{m}}_{\text{rel},\text{min}}$; as determined by 2nd Law compliance. From Eq. (2.1): for the case, $\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}=0$ (no excess payment of negentropy debt), gives ${\dot{E}}_{\text{rel},\text{min}}=\left({\dot{E}}_{\text{in}}\left./\phantom{{\dot{E}}_{\text{in}}{\mathrm{\Pi }}_{\text{E},\text{DS}}}\right){\mathrm{\Pi }}_{\text{E},\text{DS}}\right)$; similarly for ${\left(\mathrm{\Delta }{\dot{S}}_{\text{gen},\text{sur}}\right)}_{\text{m}}=0$, gives ${\dot{m}}_{\text{rel},\text{min}}=\left({\dot{m}}_{\text{in}}/{\mathrm{\Pi }}_{\text{m}}\right)$. Therefore, maximum DS-growth margins (for given in-flows) are, $\mathrm{\Delta }{\dot{E}}_{\text{grow},\text{max}}={\dot{E}}_{\text{in}}-{\dot{E}}_{\text{rel},\text{min}}={\dot{E}}_{\text{in}}·\left[1-\left(1\left./\phantom{1{\mathrm{\Pi }}_{\text{E},\text{DS}}}\right){\mathrm{\Pi }}_{\text{E},\text{DS}}\right)\right]$, $\mathrm{\Delta }{\dot{m}}_{\text{grow},\text{max}}={\dot{m}}_{\text{in}}-{\text{m}}_{\text{rel},\text{min}}={\dot{m}}_{\text{in}}·\left[1-\left(1\left./\phantom{1{\mathrm{\Pi }}_{\text{m}}}\right){\mathrm{\Pi }}_{\text{m}}\right)\right]$. In Eq. (1.1.1), corresponding maximum efficiency of using the in-flows for DS-growth are:

$$\begin{array}{cc}\hfill {\left({\eta }_{\text{grow,E}}\right)}_{max}=& 1-\left(1/{\mathrm{\Pi }}_{\text{E,DS}}\right)\text{and}{\left({\eta }_{\text{grow,m}}\right)}_{max}\hfill \\ \hfill =& 1-\left(1/{\mathrm{\Pi }}_{\text{m}}\right).\hfill \\ \hfill \end{array}$$ 2.3.1

These maximum values of efficiencies increase with higher processing of the mass-energy in-flows by DS, i.e. at higher values of ${\mathrm{\Pi }}_{\text{E},\text{DS}},{\mathrm{\Pi }}_{\text{m}}$.

Equation (2.1) has 2nd Law compliance inequality for the radiation energy exchanged, which is applicable to a plant-leaf. Following relation is obtained by re-writing $\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}$ using Eq. (1):

$$\left(s_{\text{E},\text{rel}}-s_{\text{E},\text{in}}\right)·{\dot{E}}_{\text{rel}}=\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}+s_{\text{E},\text{in}}·{\dot{E}}_{\text{DS}}>>0.$$ 2.4

Satisfying the above inequality is needed for compliance with the following inequalities separately:

• (i)Net radiation entropy generated in the surroundings, $\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}\ge 0$ [ref. Equation (2.1)], for 2nd Law compliance in the relatively infinite surroundings of DS. It is the excess negentropy debt paid for sustained DS-existence by radiation entropy generation in the surroundings.
• (ii)$s_{\text{E},\text{in}}·{\dot{E}}_{\text{DS}}$ ≥ 0, for growth of self-organised DS, by integrating low entropy density content.

Using Eq. (1.1), Eq. (2.4) reduces to,

$$s_{\text{E,rel}}-s_{\text{E,in}}=\frac{\mathrm{\Delta }{\dot{S}}_{\text{rad,gen}}}{{\dot{E}}_{\text{in}}\left(1-{\mathrm{\eta }}_{\text{grow,E}}\right)}+\frac{s_{\text{E,in}}·{\mathrm{\eta }}_{\text{grow,E}}}{1-{\mathrm{\eta }}_{\text{grow,E}}}>>0.$$ 2.4.1

Processing level of self-organized DS (e.g. plant) due to the difference, $s_{\text{E},\text{rel}}-s_{\text{E},\text{in}}$, is explained as follows:

• (i)More growth of self-organized DS is when, ${\mathrm{\eta }}_{\text{grow,E}}$ is high, but s_E,in (integrated with DS) is low. Photosynthetic efficiency of photosynthetically active radiation [= ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ , shown later in Eq. (3.1)] is ${\mathrm{\eta }}_{\text{grow},\text{E}}$.
• (ii)For given processing level [fixed left hand-side in Eq. (2.4.1)], growth of self-organized DS is by reducing $\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}$ subject to, $\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}>0$.

Motivation, objectives and scope

Plant with growth potential is a self-organized dissipative structure (DS), just as any other life-form. For its existence and growth with mandatory 2nd Law compliance [Eq. (2.1)], its leaves generate radiation entropy in the surroundings. Radiation entropy generation is by using part of the incident energy in Photosynthetically Active Radiation (PAR), which is processed by the plant-leaf but not absorbed for plant-growth. The PAR is from 0.4–0.7 µm [6], which has significant overlap with the visible spectrum, 0.38–0.75 μm. The PAR includes, λ_peak = 0.5 μm (wavelength of peak emission from Wien’s Displacement Law, within green colour), corresponding to the Sun’s surface temperature, T_sun ≈ 5772 K.

This theoretical research thermodynamically studies the photochemical reaction that all plants accomplish naturally with ease, but so far has not been achieved artificially (even at laboratory-scale). Difficulty is the formation of highly ordered C₆H₁₂O₆ molecules having low entropy density (with several covalent bonds), using energy of sunlight in PAR for radiation entropy generation in the surroundings. Radiation at high entropy density released (s_E,rel) at the plant-leaf’s temperature (to generate entropy in surroundings) relative to the low entropy density of PAR absorbed (s_E,in), is by the processing of PAR. In photosynthesis, energy in PAR is used to break the bonds in CO₂ and liquid water. Atoms are then recombined into O₂ and C₆H₁₂O₆ in which, radiation energy is stored as potential energy in the formation of covalent bonds of glucose. Hence, chemical energy stored in the bonds is a measure of glucose production. This light to chemical energy conversion in photosynthesis annually results in the world-wide storage of ~ 3 × 10²¹ J of energy in glucose, C₆H₁₂O₆. But world’s net population is increasing and economic development is diminishing the land resources available for agricultural cultivation. Therefore, doubling of agricultural productivity is needed by the end of this century for sustainability [7]. Photosynthetic efficiency [${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ , Eq. (1.1)] is a key parameter determining the thermodynamic performance of a plant-leaf. It is the fraction of radiation energy in PAR intake that is absorbed for plant-growth by producing chemical energy, which is stored in the bonds of glucose [8]. Hence, an insightful study of light-to-chemical energy conversion [9] using the thermodynamic efficiency of photosynthesis ${\mathrm{\eta }}_{\text{th}-\text{ph}}$ [mathematically defined later in Eq. (3.1.4)] and ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$, is needed. Re-visit to the application of 1st and 2nd laws can give better insights into the mechanism of photosynthesis. Energy balance equation (from 1st Law) and entropy balance equation (from 2nd Law) are strongly coupled in photosynthesis, which is modelled by combining them [Eq. (2.4.1)].

Early studies on the thermodynamics of photosynthesis were based on the Carnot cycle model [10], which concluded that ${\mathrm{\eta }}_{\text{th}-\text{ph}}$ is inherently limited. Bolton & Hall [11] theoretically estimated that the maximum thermodynamic efficiency of photosynthesis, ${\mathrm{\eta }}_{\text{th}-\text{ph},\text{max}}$  ~ 13%, in green-plants that release O₂ by inducing the oxidation of water in bright sunlight. Parson [12] argued that the Carnot cycle model does not apply to photo-chemical process, because ${\mathrm{\eta }}_{\text{th}-\text{ph}}$ is determined by the chemical kinetics of species in reaction. Jennings et al. [13] supported Parson [12] on the invalidity of Carnot cycle model to describe ${\mathrm{\eta }}_{\text{th}-\text{ph}}$, but concluded that 2nd Law is violated. This conclusion was soon refuted by Lavergne [14], who discussed a photochemical energy transducer as a model for photosynthesis that must necessarily satisfy 2nd Law. Knox & Parson [15] theoretically proved that 2nd Law is not violated in a photochemical energy conversion, regardless of the value of photosynthetic efficiency, η_ph. Literature indicates that there is a lack of conceptual clarity on the non-equilibrium thermodynamic basis of η_ph. It is addressed in this theoretical study by modelling plant as self-organising DS and by using 1st Law (conservation principle) and 2nd Law.

Plant as self-organizing dissipative structure (DS)

Incident sunlight on a plant-leaf’s surface has two parts, ${\dot{E}}_{\text{inc}}={\dot{E}}_{\text{inc},\text{PAR}}+{\dot{E}}_{\text{inc},\text{PnAR}}$; where, ${\dot{E}}_{\text{inc},\text{PnAR}}$ is Photosynthetically non-Active Radiation (PnAR). Distribution of incident sunlight in PAR on a plant-leaf is as follows [16]: ${\dot{E}}_{\text{inc},\text{PAR}}={\dot{E}}_{\mathbf{in},\mathbf{PAR}}+{\dot{E}}_{\text{refl},\text{no}\_\text{proc}}+{\dot{E}}_{\text{trans},\text{no}\_\text{proc}}=\left({\dot{E}}_{\mathbf{abs},\mathbf{grow}}+{\dot{E}}_{\mathbf{em}\_\mathbf{Tleaf}}\right)+{\dot{E}}_{\text{refl},\text{no}\_\text{proc}}+{\dot{E}}_{\text{trans},\text{no}\_\text{proc}}$. In this equation, ${\dot{E}}_{\text{in},\text{PAR}}$ is the energy intake by the plant-leaf, part of which is absorbed for plant-growth $\dot{E}_{\text{abs},\text{grow}}^{}$ and the remaining part is emitted at the leaf temperature (T_leaf) after processing, ${\dot{E}}_{\text{em}\_\text{Tleaf}}$. Plant-leaf typically takes in ~ 85% of incident sunlight in PAR, i.e. plant-leaf’s absorptivity, ${\mathrm{\alpha }}_{\text{leaf},\text{PAR}}=\left({\dot{E}}_{\text{in},\text{PAR}}\left./\phantom{{\dot{E}}_{\text{in},\text{PAR}}{\dot{E}}_{\text{inc},\text{PAR}}}\right){\dot{E}}_{\text{inc},\text{PAR}}\right)\approx 0.85$ [17]; which differs from the total absorptivity, ${\mathrm{\alpha }}_{\text{leaf},\text{tot}}=\left({\dot{E}}_{\text{in},\text{PAR}}+{\dot{E}}_{\text{abs},\text{PnAR}}\right)\left./\phantom{\left({\dot{E}}_{\text{in},\text{PAR}}+{\dot{E}}_{\text{abs},\text{PnAR}}\right)\left({\dot{E}}_{\text{inc},\text{PAR}}+{\dot{E}}_{\text{inc},\text{PnAR}}\right)}\right)\left({\dot{E}}_{\text{inc},\text{PAR}}+{\dot{E}}_{\text{inc},\text{PnAR}}\right)$. A small percentage of ${\dot{E}}_{\text{inc},\text{PAR}}$ (~ 10%) is reflected without processing (${\dot{E}}_{\text{refl},\text{no}\_\text{proc}}$ ) and the remaining 5% is transmitted without processing,${\dot{E}}_{\text{trans},\text{no}\_\text{proc}}$ . Spectral absorptivity, α_λ, is the highest for blue and red light ~ 0.80–0.95 and lower for green light ~ 0.5–0.8 [18].

By energy conservation, PAR taken in by the plant-leaf is distributed as follows:

$${\dot{E}}_{\mathbf{in},\mathbf{PAR}}={\dot{E}}_{\mathbf{abs},\mathbf{grow}}+{\dot{E}}_{\mathbf{em}\_\mathbf{Tleaf}};$$ 3

where, ${\dot{E}}_{\text{in},\text{PAR}}={\dot{E}}_{\text{in}}$, in Eq. (1.1). Part of ${\dot{E}}_{\text{in},\text{PAR}}$ (determined by photosynthetic efficiency) is absorbed for plant-growth $\dot{E}_{\text{abs},\text{grow}}^{}$ [= Ė_DS, in Eq. (1.1)] = ${\mathrm{\eta }}_{\text{ph},\text{PAR}}·{\dot{E}}_{\text{in},\text{PAR}}$. The ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ is the fraction of sunlight in PAR that is taken in by the plant-leaf and absorbed for plant-growth (by storing in covalent bonds of glucose). From Eq. (1.1),

$$\begin{array}{cc}\hfill {\eta }_{\text{grow},\text{E}}=& {\eta }_{\text{ph},\text{PAR}}=\left({\dot{E}}_{\text{abs},\text{grow}}/{\dot{E}}_{\text{in},\text{PAR}}\right)\hfill \\ \hfill =& \text{1}-\left({\dot{E}}_{\text{em}\_\text{Tleaf}}/{\dot{E}}_{\text{in},\text{PAR}}\right).\hfill \\ \hfill \end{array}$$ 3.1

Photosynthetic efficiency based on fraction of ${\dot{E}}_{\text{inc},\text{PAR}}$ used for plant-growth (differs from ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$) is given as, ${\mathrm{\eta }}_{\text{ph},\text{inc}}=\left({\dot{E}}_{\text{abs},\text{grow}}\left./\phantom{{\dot{E}}_{\text{abs},\text{grow}}{\dot{E}}_{\text{inc},\text{PAR}}}\right){\dot{E}}_{\text{inc},\text{PAR}}\right)=\left({\dot{E}}_{\text{abs},\text{grow}}\left./\phantom{{\dot{E}}_{\text{abs},\text{grow}}{\dot{E}}_{\text{in},\text{PAR}}}\right){\dot{E}}_{\text{in},\text{PAR}}\right)\left({\dot{E}}_{\text{in},\text{PAR}}\left./\phantom{{\dot{E}}_{\text{in},\text{PAR}}{\dot{E}}_{\text{inc},\text{PAR}}}\right){\dot{E}}_{\text{inc},\text{PAR}}\right)$. From Eq. (3.1), ${\mathrm{\eta }}_{\text{ph},\text{inc}}={\mathrm{\eta }}_{\text{ph},\text{PAR}}·{\mathrm{\alpha }}_{\text{leaf},\text{PAR}};\text{thus},{\mathrm{\eta }}_{\text{ph},\text{inc}}<{\mathrm{\eta }}_{\text{ph},\text{PAR}}$ (since, α_leaf,PAR ~ 0.85).

Plant is an open thermodynamic system, which exchanges radiation and matter with its surroundings; ref. Figure 1b. Endothermic reaction in a plant absorbing ${\dot{E}}_{\text{in},\text{PAR}}$ [in Eq. (3)] for the synthesis of glucose is,

$${\text{6CO}}_2+{\text{6H}}_2\text{O}+{\dot{E}}_{\text{in},\text{PAR}}\to {\text{C}}_6{\text{H}}_{12}{\text{O}}_6+{\text{6O}}_2+{\dot{E}}_{\text{em}\_\text{Tleaf}}\left(<{\dot{E}}_{\text{in},\text{PAR}}\right).$$ 3.1.1

By photosynthesis, ${\dot{E}}_{\text{in},\text{PAR}}$ is converted into chemical energy (Ė_abs,grow) stored in the bonds of glucose, C₆H₁₂O₆. Waste matter O₂ and grey body radiation Ė_{em_Tleaf} are released to the surroundings at higher entropy densities, than the intakes. Therefore, production of glucose retained by the plant for its growth correlates with the release of O₂ (at relatively higher specific entropy, s_O2) [19] and the release of grey body radiation, Ė_{em_Tleaf} (at s_E,rel >  > s_E,in). Glucose has lower specific entropy due to part of absorbed PAR at low s_E,in stored in its bonds, which is retained by the plant. In Eq. (3.1.1), Ė_{em_Tleaf} [= Ė_rel, in Eq. (1.1)], is thermal radiation at T_leaf, i.e. it includes PAR and is mostly in PnAR (mostly at wavelengths longer than PAR). It is distributed as grey-body radiation over the full electromagnetic spectrum (λ = 0 − ∞ μm) as per Planck’s Law:

$${\dot{E}}_{\text{em}\_\text{Tleaf}}=\left(1-{\mathrm{\eta }}_{\text{ph},\text{PAR}}\right)·{\dot{E}}_{\text{in},\text{PAR}}=A_{\text{leaf}}·{ϵ}_{\text{leaf}}·\mathrm{\sigma }·T_{\text{leaf}}^4$$ 3.1.2

In Eq. (3.1.2):

• (i)Ė_{em_Tleaf} (< ${\dot{E}}_{\text{in},\text{PAR}}$) is the re-emission of PAR processed by plant-leaf at s_E,rel (> > s_E,in);
• (ii)A_leaf is the surface area of plant-leaf;
• (iii)ε_leaf is the emissivity of plant-leaf’s surface, which is high (> 0.9) for semi-transparent fresh leaves that have water content [16];
• (iv)σ (= 5.6704 × 10⁻⁸ W/m² K⁴) is the Stephan-Boltzmann constant.

2nd law compliance in photosynthesis

From Eq. (2.3.1), the upper limit for plant-growth based on processing of PAR and 2nd Law compliance is given by the maximum photosynthetic efficiency,

$${\mathrm{\eta }}_{\text{ph},\text{PAR}-\text{max}}={\left({\mathrm{\eta }}_{\text{grow},\text{E}}\right)}_{\text{max}}=1-\left(1/{\mathrm{\Pi }}_{\text{E},\text{leaf}}\right)<1.$$ 3.1.3

At ${\mathrm{\eta }}_{\text{ph},\text{PAR}}={\mathrm{\eta }}_{\text{ph},\text{PAR}-\text{max}},{\dot{E}}_{\text{em}\_\text{Tleaf}}={\dot{E}}_{\text{em}\_\text{Tleaf},\text{min}}$, i.e. ${\dot{E}}_{\text{abs},\text{grow}}={\dot{E}}_{\text{abs},\text{grow}\_\text{max}}$ for given ${\dot{E}}_{\text{in},\text{PAR}}$. Processing level of PAR by plant-leaf is, ${\mathrm{\Pi }}_{\text{E},\text{leaf}}=\left(s_{\text{E},\text{rel}}\left./\phantom{s_{\text{E},\text{in}}{s}_{\text{E},\text{rel}}}\right)s_{\text{E},\text{in}}\right)$, ref. Eq. (2.2); therefore, for higher η_ph,PAR-max (more self-organization with growth), Π_E,leaf must be increased. Equation (3.1.3) gives the upper limit for DS-growth, (η_grow,E)_max [Eq. (2.3.1)], based on 2nd Law compliance. Thus, photosynthetic efficiency, ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$, is inherently limited by the plant-leaf’s ability to process PAR for generating radiation entropy in the surroundings.

From 2nd Law analyses of photosynthesis [20], plant physiologists find ways to increase ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ for given ${\mathrm{\eta }}_{\text{th}-\text{ph}}$ (= thermodynamic efficiency of photosynthesis) [21]. It is defined as, ${\mathrm{\eta }}_{\text{th}-\text{ph}}$ = (chemical energy stored in bonds of C₆H₁₂O₆)/[energy of PAR intake that is absorbed for growth, ${\dot{E}}_{\text{abs},\text{grow}}\left(={\mathrm{\eta }}_{\text{ph},\text{PAR}}·{\dot{E}}_{\text{in},\text{PAR}}\right)$]. Therefore,

$$\left({\text{chemical energy stored in bonds of C}}_6{\text{H}}_{12}{\text{O}}_6\right)=({\mathrm{\eta }}_{\text{ph},\text{PAR}}·{\dot{E}}_{\text{in},\text{PAR}})·{\mathrm{\eta }}_{\text{th}-\text{ph}}.$$ 3.1.4

Part of, Ė_abs,grow (= ${\mathrm{\eta }}_{\text{ph},\text{PAR}}·{\dot{E}}_{\text{in},\text{PAR}}$) is stored in the bonds of C₆H₁₂O₆ [22], which when broken releases free energy (used by plant for its growth, to do work e.g. against Earth’s gravity). From Eq. (3.1.4), increase in chemical energy stored in the bonds of C₆H₁₂O₆, i.e. more sugar production is achieved by increasing ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ (for given ${\mathrm{\eta }}_{\text{th}-\text{ph}}$). Using Eq. (2.4) for the photochemical reaction given by Eq. (3.1.1),

$$\begin{array}{cc}& \left(s_{\text{E},\text{rel}}-s_{\text{E},\text{in}}\right)·{\dot{E}}_{\text{em}\_\text{Tleaf}}\hfill \\ \hfill & =\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}+s_{\text{E},\text{in}}·{\dot{E}}_{\text{abs},\text{grow}}>>0.\hfill \end{array}$$ 3.2

Following from above are the two additional inequalities: (i) for self-organized plant to grow, s_E,in⋅Ė_abs,grow > 0; (ii) for mandatory 2nd Law compliance, net radiation entropy generated by the plant-leaf in its surroundings, $\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}$ > 0. Processing of PAR by the plant-leaf (s_E,rel >  > s_E,in) generates radiation entropy in the surroundings, $\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}$, for the mandatory 2nd Law compliance. It is the available path of least resistance in LMEP [1] (least action path in dissipative system [3]), to maximise entropy production in the surroundings.

Equation (3.2) is re-written as,

$$s_{\text{E},\text{rel}}-s_{\text{E},\text{in}}=\frac{\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}}{{\dot{E}}_{\text{em}\_\text{Tleaf}}}+s_{\text{E},\text{in}}·\frac{{\mathrm{\eta }}_{\text{ph},\text{PAR}}}{1-{\mathrm{\eta }}_{\text{ph},\text{PAR}}}>>0.$$ 3.2.1

Intensity of sunlight in PAR reduces enroute from the Sun’s surface to Earth’s surface (Ė_in,PAR), but its high quality as measured by its low entropy density (low s_E,in) remains unchanged. High entropy density difference on the left-hand side is negentropy build-up of plant [shown later in Eq. (4.2.1)], due to processing of PAR. From Eq. (3.2), dimensionless net radiation entropy generated by the plant-leaf in its surroundings is given as,

$$\begin{array}{cc}\hfill {\mathrm{\Pi }}_{\text{Srad},\text{gen}}& =\frac{\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}}{s_{\text{E},\text{in}}·{\dot{E}}_{\text{abs},\text{grow}}}\hfill \\ \hfill & =({\mathrm{\Pi }}_{\text{E},\text{leaf}}-1)·[(1/{\mathrm{\eta }}_{\text{ph},\text{PAR}})-1]-1\ge 0.\hfill \end{array}$$ 3.2.2

Net radiation entropy generated in the surroundings per unit radiation entropy absorbed for plant-growth is, ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}$  ≥ 0, for mandatory 2nd Law compliance regardless of the value of ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$. From Eq. (3.2.2), with increase in ${\mathrm{\Pi }}_{\text{E},\text{leaf}}$ (processing level), ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}$ increases for given ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$. Figure 2a shows, ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}=f\left({\mathrm{\Pi }}_{\text{E},\text{leaf}},{\mathrm{\eta }}_{\text{ph},\text{PAR}}\right)$, which decreases with increasing ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$, for given Π_E,leaf (from 18 to 30). Therefore, plant-growth is by reducing net radiation entropy generated [ref. Equation (2.4.1)] subject to 2nd Law compliance, i.e. region above, ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}$  ≥ 0 in Fig. 2a, is feasible. Rearranging Eq. (3.2.2),

$$\begin{array}{cc}\hfill {\mathrm{\eta }}_{\text{ph},\text{PAR}}=& [1-(1/{\mathrm{\Pi }}_{\text{E,leaf}})]/[1+({\mathrm{\Pi }}_{Srad,gen}/{\mathrm{\Pi }}_{E,leaf})]\hfill \\ \hfill & <{\mathrm{\eta }}_{\text{ph},\text{PAR}-\text{max}},\hfill \end{array}$$ 3.2.3

thus, ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ increases with increasing Π_E,leaf, for given Π_Srad,gen. Relation between Π_Srad,gen and ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$, has the following two limiting cases [ref. block diagram in Fig. 2b]:

• (i)Upper limit to plant-growth (zero net radiation entropy generation in surroundings) is the intercept on ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ (X) axis, ${\mathrm{\eta }}_{\text{ph},\text{PAR}}\left({\mathrm{\Pi }}_{\text{Srad},\text{gen}\_\text{min}}=0\right)={\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}$. It is given by Eq. (3.1.3), which is a special case of Eq. (3.2.3).
• (ii)Maximum net radiation entropy generation in the surroundings (no plant-growth) is for, ${\mathrm{\Pi }}_{\text{Srad},\text{gen}\_\text{max}}\to \infty$.

Fig. 2.

Dimensionless radiation entropy generated by the plant-leaf, ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}=f({\mathrm{\eta }}_{\text{ph},\text{PAR}},{\mathrm{\Pi }}_{\text{E},\text{leaf}})$. a 2-D plot of ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}=f({\mathrm{\eta }}_{\text{ph},\text{PAR}})$, for ${\mathrm{\Pi }}_{\text{E},\text{leaf}}$ from 18 to 30. b Block diagram illustrating limiting cases of ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}$ and ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$

Leaf contributes to plant-growth (${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ > 0) when, ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}$ is finite; which is characterised by high processing level of PAR (high Π_E,leaf). Plant-leaf decays (${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ < 0) when, ${\mathrm{\Pi }}_{\text{Srad},\text{gen}}<0$, after losing its ability to process PAR (low Π_E,leaf).

Heat is generated in the plant-leaf by −

• (i)processing of Ė_in,PAR in the plant-leaf due to metabolism, which coverts it in to chemical energy stored in the bonds of C₆H₁₂O₆;
• (ii)absorption of thermal radiation (primarily in infrared spectrum) at wavelengths longer than PAR [23].

For non-transpirating plant-leaf [24], T_leaf is a bit higher than the local ambient temperature (T_a) and (T_leaf − T_a) ≠ f(T_a), i.e. T_leaf increases linearly with T_a [25]. Heat loss from plant-leaf to its surroundings by convection and radiation is at rate (which lowers T_leaf & reduces disorder content in leaf): $q_{\text{HT}}=A_{\text{leaf}}·[h·(T_{\text{leaf}}-T_{\text{a}})+{ϵ}_{\text{leaf}}·\mathrm{\sigma }\left(T_{\text{leaf}}^4-T_{\text{a}}^4\right)$]. Total irreversible loss in processing incident light in PAR can be as high as 97% [26] and it increases with T_leaf (due to reduction in processing ability of plant-leaf).

Negentropy of plant as dissipative structure (DS) generating radiation entropy

Link between information collected by a system from its surroundings for sustained existence was defined as the negative of its entropy reduction by Shannon [27]. Energy alone cannot sustain dissipative structure (DS), ref. Schroedinger [28] (1945), who noted that life maintains its ordered state by “feeding” on negative entropy (negentropy), by taking order from its environment and dissipating disorder. Elitzur [29] noted that an organism (e.g. plant) must have order relative to its environment, by processing information gained from its surroundings.

Self-organization (as measured by negentropy) in plant as DS is by the continuous net in-flux of negative radiation entropy density. Processing of energy and/or mass for the release to surroundings at much higher entropy density than the in-flows, Π_E,DS >> 1 [Eq. (2.2)], Π_E,leaf >> 1 [Eq. (3.1.3)], is the net in-flux of negative entropy density in a plant as a self-organizing DS. Net in-flux of negative entropy density is needed to at least balance the rise in entropy that inevitably accompanies the irreversible processing inside the plant and also for increasing its self-organization. Mahulikar & Herwig [30] re-defined Schroedinger’s [28] notion of negentropy (s_neg) as the specific entropy contrast of DS relative to its release (immediate surroundings of DS):

$$s_{\text{neg}}=s_{\text{DS}}-s_{\text{rel}};\text{where},s_{\text{DS}}<<s_{\text{rel}}\Rightarrow s_{\text{neg}}<<0.$$ 4

From Eq. (2.4.1), mass-energy content of a self-organized DS (s_DS) is made up of processed in-flows with low entropy density (s_E,in, s_in), because higher entropy density content (s_rel) is released to the surroundings. From Eqs. (3.1.1) and (3.2.2), higher entropy density mass-energy content (O₂, E_{em_Tleaf}) is released; thus,

$$s_{\text{in}}\to s_{\text{DS}}<<s_{\text{rel}},\text{and}{s}_{\text{E},\text{in}}\to s_{\text{E},\text{DS}}<<s_{\text{E},\text{rel}}.$$ 4.1

Negentropy production (for self-organization) and its link to possible growth of DS (e.g. plant) is a special case of LMEP [1], applicable to DS existence. Formation of complex glucose molecules with large amount of information (large number of bonds) contributes to the negentropy build-up of plant, referred as negative entropy production [13]. Negentropy build-up by the decrease in entropy in building C₆H₁₂O₆ molecules [31] in the plant is made possible by the simultaneous release of Ė_{em_Tleaf} [Eq. (3.1.2)] at high s_E,rel to surroundings (compensation for mandatory 2nd Law compliance).

Specific entropy per unit mass, s_DS (entropy-density of mass, SI units [J/kg-K]), relates with entropy-by-energy ratio, s_E,DS (entropy-density of energy, SI units [K⁻¹]) as; s_E,DS = (s_DS/c²), where, c (= 2.998 × 10⁸ m/s) is the speed of light. From Eq. (4), negentropy of plant as DS, which exists as matter by the exchange of radiation energy and matter with surroundings is,

$$\begin{array}{cc}\hfill s_{\text{neg}}& =s_{\text{neg},\text{rad}}+s_{\text{neg},\text{mat}}=c^2·\left(s_{\text{E},\text{DS}}-s_{\text{E},\text{rel}}\right)\hfill \\ \hfill & +\left(s_{\text{DS}}-s_{\text{rel}}\right)\hfill \\ \hfill & =c^2·s_{\text{E},\text{DS}}·\left[1-\left(s_{\text{E},\text{rel}}\left./\phantom{s_{\text{E},\text{rel}}{s}_{\text{E},\text{DS}}}\right)s_{\text{E},\text{DS}}\right)\right]+s_{\text{DS}}·\hfill \\ \hfill & \left[1-\left(s_{\text{rel}}\left./\phantom{s_{\text{rel}}{s}_{\text{DS}}}\right)s_{\text{DS}}\right)\right]<<0.\hfill \\ \hfill \end{array}$$ 4.2

The s_neg (SI units [J/kg−K]) of plant as DS existing as matter is defined based on specific entropy (entropy density per unit mass of matter). Therefore, the change in entropy density of radiation energy exchanged (entropy per unit energy), is multiplied by, c². In the right-hand side of Eq. (4.2):

• (i)s_neg,rad is negentropy build-up of the plant as DS, by the generation of radiation entropy due to release of Ė_{em_Tleaf} at s_E,rel.
• (ii)s_neg,mat is negentropy build-up of the plant as DS, by matter exchanged.
• (iii)Ratios of entropy densities, (s_E,rel/s_E,DS) and (s_rel/s_DS), correlate with negentropy build-up of DS, for given s_E,DS and s_DS.

Plants can build their s_neg also by processing radiation (in addition to matter); where, s_neg,rad >> s_neg,mat; hence, this study focuses exclusively on s_neg,rad. Equation (4.2) is written for radiation energy exchanged as [using approximation given by Eq. (4.1)],

$$s_{\text{neg},\text{rad}}=c^2·s_{\text{E},\text{DS}}·[1-\left(s_{\text{E},\text{rel}}/s_{\text{E},\text{DS}}\right)]\approx c^2·s_{\text{E},\text{DS}}·(1-{\mathrm{\Pi }}_{\text{E}},\text{leaf})\approx -c^{\mathbf{2}}·(s_{\mathbf{E},\mathbf{rel}}-s_{\mathbf{E},\mathbf{in}})<<\mathbf{0}.$$ 4.2.1

Change in entropy density of radiation from low s_E,in to high s_E,rel by the leaf is amplified by a huge factor (c²), while determining the negentropy build-up of the plant. Therefore, even a small increase in s_E,rel and a small reduction in s_E,in can significantly increase the negentropy build-up of the plant. It is difficult to artificially change the entropy density of radiation (also at laboratory-scale), as is done by a plant-leaf. From Eq. (4.2.1), higher negentropy build-up of plant is by increasing the processing level of absorbed PAR, i.e. by increasing Π_E,leaf. Possible role of transpiration cooling of plant-leaf in the entropy production in surroundings [32] is not considered in this study. Using Eq. (3.2.1), the magnitude of negentropy build-up of plant is given as,

$$\left(|s_{\text{neg},\text{rad}}|/c^2\right)=\frac{\mathrm{\Delta }{\dot{S}}_{\text{rad},\text{gen}}}{{\dot{E}}_{\text{em}\_\text{Tleaf}}}+s_{\text{E},\text{in}}·\frac{{\mathrm{\eta }}_{\text{ph},\text{PAR}}}{1-{\mathrm{\eta }}_{\text{ph},\text{PAR}}}.$$ 4.2.2

Part of this negentropy build-up can be made available as the source of free energy of the plant, as per the interpretation of Schroedinger [28]. This part can be stored as plant-growth and then made available as free-energy to the plant (e.g. to do work against Earth’s gravity), which is 2nd term in Eq. (4.2.2). Remaining part of $\left|s_{\text{neg},\text{rad}}\right|$ is dissipated in the surroundings [1st term in Eq. (4.2.2)], which is the excess payment of negentropy debt for existence of plant as self-organized DS. Thus, Eq. (4.2.2) gives negentropy build-up of the plant as sum of:

• (i)Net radiation entropy generated by the plant-leaf (per unit radiation energy released) in its surroundings, which is the mandatory compliance of 2nd Law.
• (ii)Possible plant-growth through the photochemical reaction given by Eq. (3.1.1), by integrating low entropy density PAR in the bonds of C₆H₁₂O₆.

In Eq. (4.2.1), s_E,in of the absorbed PAR is given as [33], s_E,in = $\frac{S_{\mathrm{\nu }}}{E_{\mathrm{\nu }}}$ = $\frac{k}{h·\nu }·\left[\left(1+\frac{1}{{\mathrm{\xi }}_{\text{sun}}}\right)·\text{ln}\left(1+{\mathrm{\xi }}_{\text{sun}}\right)-\text{ln}\left({\mathrm{\xi }}_{\text{sun}}\right)\right]$; where, ${\xi }_{\text{sun}}={\left[\text{exp}\left(\frac{h·\nu }{k·T_{\text{sun}}}\right)-1\right]}^{-1}$. If Λ_sun(T_sun) = [h⋅c/(k⋅T_sun)] ≈ 2.492746 µm (dimensional constant, h: Planck’s const., k: Boltzmann’s const.),

$$\begin{array}{cc}\hfill s_{\text{E,in}}\left(\lambda \right)=& \frac{\left(\lambda /{\mathrm{\Lambda }}_{\text{sun}}\right)}{T_{\text{sun}}}·\left\{exp\left(\frac{{\mathrm{\Lambda }}_{\text{sun}}}{\lambda }\right)·ln\left[\frac{1}{1-exp\left(-\frac{{\mathrm{\Lambda }}_{\text{sun}}}{\lambda }\right)}\right]\right)\hfill \\ \hfill & \left(-ln\left[\frac{1}{exp\left(\frac{{\mathrm{\Lambda }}_{\text{sun}}}{\lambda }\right)-1}\right]\right\};\hfill \\ \hfill \end{array}$$ 4.3

where, (Λ_sun/λ) > 1, in PAR. Spectral temperature of Sun {T_sunλin [K], Fig. 3a} is the reciprocal of its spectral entropy-density,

$$T_{sun\lambda }(\lambda )=[1/s_{\text{E},\text{in}}(\lambda )].$$ 4.3.1

Fig. 3.

Variation of entropy density of radiation, s_E. a Photosynthetically Active Radiation (PAR) absorbed by plant-leaf at,$T_{\text{sun}\lambda }=1/s_{\text{E},\text{in}}$. b Grey-body emission by plant-leaf as its release to the surroundings

It represents the Sun’s temperature at a particular λ, for given surface temperature, T_sun (≈ 5772 K). As shown in Fig. 3a, T_sunλ decreases with increasing λ (almost linearly) from 4974.5 to 4520.7 K (ΔT_sunλ ~ 9.123%), in PAR. In Eq. (4.2.1), s_E,rel of the released spectrum [distributed as black-body radiation at T_leaf (in [K])] is,

$$s_{\text{E},\text{rel}}\left(T_{\text{leaf}}\right)=\frac{4}{3}\frac{\left(1-{\mathrm{\eta }}_{\text{ph},\text{PAR}}\right)·n·h·\nu }{T_{\text{leaf}}}·\frac{1}{\left(1-{\mathrm{\eta }}_{\text{ph},\text{PAR}}\right)·n·h·\nu }=\frac{4}{3T_{\text{leaf}}}$$ 4.4

Thus, ratio of the entropic potential of grey-body radiation (emitted at T_leaf) by its energy is inversely proportional to T_leaf (also inferred from [34]). Figure 3b gives s_E,rel(T_leaf) [>> s_E,in(λ), ref. Eq. (4.1)], which shows that s_E,rel decreases monotonically from 5.98 × 10⁻³ to 4.128 × 10⁻³ K⁻¹ (− $\mathrm{\Delta }{s}_{\text{E},\text{rel}}$ ~ 30.97%), with increasing T_leaf [= f(T_a)] in the range, − 45 to + 55 °C. Therefore, change in T_a has a greater role in the plant’s ${\mathrm{\Pi }}_{\text{E},\text{leaf}}$, ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}$, and $|s_{\text{neg},\text{rad}}|$, relative to λ of sunlight processed in PAR.

From Eqs. (4.3.1) and (4.4), the ratio,

$$\begin{array}{cc}\hfill {\mathrm{\Pi }}_{\text{E},\text{leaf}}\left(T_{\text{leaf}},\lambda \right)=& \left[s_{\text{E},\text{rel}}\left(T_{\text{leaf}}\right)/s_{\text{E},\text{in}}(\lambda )\right]\hfill \\ \hfill =& \left(4/3\right)\left(T_{\text{sun}\lambda }/T_{\text{leaf}}\right);\hfill \\ \hfill \end{array}$$ 4.4.1

is in Figs. 4a, b. It shows that, s_E,rel >> s_E,in (Π_E,leaf ~ 20), which leads to the negentropy build-up of the plant as self-organizing DS [ref. Eq. (4.2.1)]. From the inequality in Eq. (3.2.1), Π_E,leaf (>> 1) gives enough margin for the sustenance and negentropy build-up of plant, with additional possibility of growth (limited by η_{ph,PAR_max}). From Fig. 4, Π_E,leaf is seen to be a stronger function of T_leaf than λ in PAR and from Eq. (4.4.1), Π_E,leaf ∝ (1/T_leaf) for given λ. Thus,

• (i)Higher processing level of the plant-leaf is of blue colour, followed by green and then red colour. Balegh & Biddulph [35] showed that green light once absorbed drives photosynthesis with higher ${\mathrm{\eta }}_{\text{ph}}$ than red-light. This reported observation agrees with the thermodynamic basis of increasing Π_E,leaf with decreasing λ, shown in Fig. 4a, i.e. green-light has higher Π_E,leaf than red (& lower than blue-light). Measured absorptance spectra showed that ordinary green leaves of land plants absorb a substantial fraction of green light [36].
• (ii)Higher processing level of the plant-leaf is at lower T_leaf. Lower T_leaf (at lower ambient temperature, T_a, or by transpiration cooling of plant-leaf for given T_a) facilitates the processing of PAR. That the spectral quantum yield increases with decreasing T_leaf is also reported by McCree [6], based on experimental studies on oat leaf.

Fig. 4.

Plant-leaf’s processing level of PAR, ${\mathrm{\Pi }}_{\text{E},\text{leaf}}=f(T_{\text{leaf}},\lambda )$. a 2-D plot of ${\mathrm{\Pi }}_{\text{E},\text{leaf}}\left(T_{\text{leaf}}\right)$ for different λ. b 3-D plot of ${\mathrm{\Pi }}_{\text{E},\text{leaf}}\left(T_{\text{leaf}},\lambda \right)$

Figure 5a shows that ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}=f({\mathrm{\Pi }}_{\text{E},\text{leaf}})$ [Eq. (3.1.3)], increases monotonically from 0 to 0.967 with Π_E,leaf increasing from 1 to 30. It also shows the range of Π_E,leaf in this study from 18.39 to 29.12, which corresponds to η_{ph,PAR_max} varying from, 0.945584—0.965625 (ref. Table 2). Qualitative trends of ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}=f(T_{\text{leaf}},\lambda )$ in Figs. 5b,c are similar to those of Π_E,leaf(T_leaf,λ) in Figs. 4a,b. From Eqs. (3.1.3), (4.3.1), and (4.4), linear relation between η_{ph,PAR_max} and T_leaf is obtained as;

$${\mathrm{\eta }}_{\text{ph,PAR\_max}}=1-\left(3\left./\phantom{34}\right)4\right)·\left(T_{\text{leaf}}\left./\phantom{T_{\text{leaf}}{T}_{\text{sunl}}}\right)T_{\text{sun}\lambda }\right).$$ 4.5

Fig. 5.

Maximum photosynthetic efficiency, ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}$. a Dependence on processing level of PAR, ${\mathrm{\Pi }}_{\text{E},\text{leaf}}$. b 2-D plot of ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}=f\left(T_{\text{leaf}}\right)$ for different λ in PAR. c 3-D plot of ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}=f\left(T_{\text{leaf}},\lambda \right)$

Table 2.

Highest and lowest thermodynamic performance of plant-leaf in PAR in feasible T_leaf-range

Performance parameter (Φ)	${\mathrm{\Pi }}_{\text{E},\text{leaf}}$	$\left|s_{\text{neg},\text{rad}}\right|\left./\phantom{\left|s_{\text{neg},\text{rad}}\right|c^2}\right)c^2$	${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}$
Tleaf =  + 55 °C, λ = 0.7 μm; Lowest Performance:	18.39	3.844 × 10−3 K−1	0.945584
Tleaf =  − 45 °C, λ = 0.4 μm; Highest Performance:	29.12	5.647 × 10−3 K−1	0.965625
[(Φmax − Φmin)/Φmin] × 100%	58.35	46.90	2.119

With increasing T_leaf [for given T_sunλ(λ)], η_{ph,PAR_max} decreases linearly; and for increasing λ in PAR from 0.4 to 0.7 μm, η_{ph,PAR_max} decreases [for given T_leaf, since, T_sunλ(λ) decreases]. Therefore, upper limit to plant-growth is raised at lower T_leaf and by the leaf’s ability to process lower λ in PAR. Rate of sugar production correlates with increase in plant-growth [limited by η_{ph,PAR_max} in Eq. (4.5)], which is higher at low T_leaf [24]. Results in Figs. 4, 5 are qualitatively (in trend) analogous to the Carnot cycle model [37]. Maximum thermodynamic cycle efficiency is that of a reversible Carnot engine, which does not generate net entropy in the surroundings. Efficiency of ideal Carnot cycle operating between heat source at absolute temperature, T_source, and heat sink at absolute temperature, T_sink, is given as;

$${\mathrm{\eta }}_{\text{cycle}\_\text{max}}=1-\left(T_{\text{sink}}/T_{\text{source}}\right).$$ 4.5.1

Comparing Eqs. (4.5) and (4.5.1): (i) T_sunλ is analogous to T_source; (ii) T_leaf is analogous to T_sink.

From Eqs. (4.2.1), (4.3.1), and (4.4),

$$\begin{array}{cc}& s_{\text{neg},\text{rad}}=-c^2\times \left(\frac{4}{3T_{\text{leaf}}}-\frac{1}{T_{\text{sun}\lambda }}\right);\text{i}.\text{e}.\hfill \\ \hfill & \left(\left|s_{\text{neg},\text{rad}}\right|/c^2\right)=\left(\frac{4}{3T_{\text{leaf}}}-\frac{1}{T_{\text{sun}\lambda }}\right).\hfill \end{array}$$ 4.6

Negentropy build-up of the plant as dissipative structure (DS) is determined by the difference between Sun’s spectral temperature (T_sunλ) and its leaf’s temperature, T_leaf. Figure 6 give the variation of ($\left|s_{\text{neg},\text{rad}}\right|$/c²) with T_leaf and λ in PAR as per Eq. (4.6), which shows its increase with decreasing T_leaf and weak dependence on λ (though $\left|s_{\text{neg},\text{rad}}\right|$ is somewhat higher at lower λ). From Eqs. (4.5) and (4.6), relation between maximum plant-growth and negentropy build-up is obtained as,

$${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}=\frac{T_{\text{sun}\lambda }}{T_{\text{sun}\lambda }+{\left(\frac{\left|s_{\text{neg},\text{rad}}\right|}{c^2}\right)}^{-1}}$$ 4.6.1

Fig. 6.

Variation of $\left(\left|s_{\text{neg},\text{rad}}\right|/c^2\right)$ with T_leaf and λ in PAR. a 2-D plot of $\left(\left|s_{\text{neg},\text{rad}}\right|/c^2\right)=f\left(T_{\text{leaf}}\right)$ for different λ. b 3-D plot of $\left(\left|s_{\text{neg},\text{rad}}\right|/c^2\right)=f\left(T_{\text{leaf}},\lambda \right)$

Since T_sunλ is fixed, increase in ($\left|s_{\text{neg},\text{rad}}\right|$/c²) results in increase in η_{ph,PAR_max}, both are higher at lower T_leaf. For the range, T_leaf =  − 45 to 55 °C and λ = 0.4–0.7 μm, highest and lowest performance parameters are in Table 2, which are the diagonally opposite corners in Figs. 4b, 5c, and 6b. Increase from lowest to highest is the most for Π_E,leaf (processing level, 58.35%); followed by $\left|s_{\text{neg},\text{rad}}\right|$ (negentropy build-up, 46.9%); and then η_{ph,PAR_max} (maximum growth potential, 2.119%). These quantitative estimates highlight the dominant purpose of processing of PAR by leaf for radiation entropy generation in the surroundings for mandatory 2nd Law compliance, rather than for possible plant-growth.

Conclusions

• (i)Plant functions as a self-organized dissipative structure (DS), because its leaf generates net radiation entropy in the surroundings for existence of plant with possible growth. Radiation entropy production in the surroundings by plant-leaf is requirement for mandatory 2nd Law compliance, for the photochemical reaction to take place leading to the formation of glucose.
• (ii)Radiation entropy production is a simple task for any green plant but is not yet realised artificially (even at laboratory-scale), due to the difficulty in achieving the high processing level of PAR. Entropy density (entropy-energy ratio) of Photosynthetically Active Radiation (PAR) that is processed and then emitted by the plant-leaf (s_E,rel) exceeds that absorbed (s_E,in), by a factor, Π_E,leaf = (s_E,rel/s_E,in) ~ 20.
• (iii)The Π_E,leaf >> 1, is an entropic measure of processing level of PAR by the plant-leaf. It determines the margin for plant-growth [Eq. (3.2.2)] based on photosynthetic efficiency in PAR, ${\mathrm{\eta }}_{\text{ph},\text{PAR}}$ (> 0, for plant-growth). The Π_E,leaf is a stronger function of the leaf’s temperature (higher at lower T_leaf) relative to the wavelength within PAR processed (higher at lower λ).
• (iv)Dimensionless net radiation entropy generated by the plant-leaf in its surroundings, Π_Srad,gen, is introduced; where, Π_Srad,gen > 0, must be satisfied for mandatory 2nd Law compliance. The Π_Srad,gen is the net radiation entropy generated in surroundings per unit radiation entropy absorbed for plant-growth. For, Π_Srad,gen = 0, plant-growth is maximum; and for, ${\mathrm{\Pi }}_{\text{Srad,gen}}\to \infty$, there is no plant-growth.
• (v)Plants can build their negentropy (s_neg < 0) also by processing radiation (in addition to matter), which has a much higher contribution to their self-organization with growth potential. In the negentropy build-up of plant by radiation entropy generation (s_neg,rad < 0), changes in s_E,rel and s_E,in are multiplied by a huge factor, c²; where, c is the speed of light. Hence, even a small rise in s_E,rel and a small drop in s_E,in can significantly increase the magnitude of the negentropy build-up of plant by radiation entropy generation, $\left|s_{\text{neg},\text{rad}}\right|$; which increases with decreasing T_leaf and is higher at lower wavelength (in PAR) processed.
• (vi)Part of $\left|s_{\text{neg},\text{rad}}\right|$ can be the source of free energy of plant; since, it can be stored as plant-growth. Remaining part of $\left|s_{\text{neg},\text{rad}}\right|$ is dissipated in the surroundings, which is the excess payment of negentropy debt as part of mandatory 2nd Law compliance for the existence of plant as self-organized DS.
• (vii)The η_{ph,PAR_max} is obtained [Eq. (4.5)] in terms of T_leaf and the spectral temperature of the Sun, $T_{\text{sun}\lambda }[=(1/s_{\text{E},\text{in}})]$. Expression for ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}$ correlates with that for the maximum thermodynamic efficiency of reversible Carnot engine, ${\mathrm{\eta }}_{\text{cycle}\_\text{max}}$. For both, ${\mathrm{\eta }}_{\text{ph},\text{PAR}\_\text{max}}$ and ${\mathrm{\eta }}_{\text{cycle}\_\text{max}}$, no net entropy is generated in the surroundings.

Author contributions

(1) Shripad P. Mahulikar—Conceived all the presented original ideas, developed the theory, performed the analysis and verified the analytical methods, wrote the manuscript with inputs and support from (2) and (3). (2) Pallavi Rastogi—Generated the results with guidance of (1), contributed to the interpretation and discussion of results, provided critical feedback. (3) Aitor Erkoreka—Provided critical feedback and helped shape the research, analysis and manuscript content.

Funding

Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. HORIZON EUROPE Marie Sklodowska-Curie Actions, HORIZON-MSCA-2022-COFUND-101126600-SmartBRAIN3, Shripad P. Mahulikar.

Data availability

This manuscript has no associated data. [Authors’ comment: This is a theoretical study, and no experimental data has been listed.]