Theory
A brief description of the Avrami’s model we refer to is reported in Appendix 3.
Instead, the logistic function adopted in this work is defined by Eq. (1), proposed in [24, 29]:
$$ X(t) = \frac{{A_{{\text{c}}} }}{{A_{{\text{t}}} }} \cdot \left( {\frac{1}{{1 + \exp^{{ - r(t - t_{{\text{p}}} )}} }}} \right) $$
(1)
where, microalgae coverage X (–) is a function of time t (day) and the three parameters, namely Ac/At, r and tp, can be defined from experimental measures. In particular, the first one is the maximum covered area ratio (–), being Ac the area covered by microalgae and At the total area of the sample. It represents the horizontal asymptote ranging between 0 and 1. The r parameter (day−1) can be defined as the intrinsic growth rate [24] while the tp parameter (day) is defined as the inflection point of the growth curve and it is the day in which microalgae coverage (Ac/At)/2 is reached. In this work both r and tp are calculated through iterations by minimizing the least squares value between experimental data and calculated values [24]. In particular, according to such method, the two parameters were calculated through iteration as a pair of values (tp, r) that minimizes the sum of the squares of the residuals. Residuals were considered as the differences between the experimental values \(X_{{\exp_{m,i} }}\) and the ones obtained with the logistic equation \(X(t_{{\text{p}}} ,r)_{i}\) for each measuring time [30], as reported in Eq. 1a.
$$ \left( {t_{{\text{p}}} ,r} \right):\min \left( {\sum\limits_{t = i} {\left( {X_{{\exp_{m,i} }} - X(t_{{\text{p}}} ,r)_{i} } \right)^{2} } } \right) $$
(1a)
Moreover, the model first derivative is always higher than 0 for every time values: hence, no decreasing trend can be observed as happening for the Avrami’s equation (see Sect. 3.2).
It is important to underline that in the following sections Eq. 1 is compared to equation C1:
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from time t = 0 to the time of the last experimental measure (condition 1);
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by considering the coverage equal to zero before the latency time, for taking into account the physical aspects involved in its formulation [17] (condition 2).
Experimental tested materials
Previous experimental microalgae growth data, already modelled by the (modified) Avrami’s model, on fired bricks, sandstones and limestones [13,14,15,16] are selected, as reported in Table 1. The relative Avrami’s curves are thus collected from such references, while the logistic ones are determined from the beginning for all the materials (Table 1).
Table 1 List of the porous building materials considering substrate properties, temperature and surface treatments [13,14,15,16] Methods for the comparison
Overlapping the experimental data
The first comparison involves the assessment of which model could better overlap the experimental data. To assess that, the comparison is run generally evaluating: how many times the models overlap the data and their fitting quality. Concerning the values out per each model, this work evaluates when they are out according to each growth phase and how far from the experimental data they are.
For the first comparison, this works determines the percentage of values that validates condition (2):
$$ \min (X_{{\exp_{1} }} ,...,X_{{\exp_{3} }} )_{i} \le X(t = i) \le \max (X_{{\exp_{1} }} ,...,X_{{\exp_{3} }} )_{i} $$
(2)
where X(t = i) is the calculated covered area for both the models at the ith time, that is, the time (days) when the measure was made during the microalgae growth (e.g.: 0, 7 days, 14 days, …, 70 days) and Xexp1, …, Xexp3 correspond to the experimental measures of the 3 samples respectively.
To the same aim, a comparison between the fitting quality index R% (–) of the two models is run. This index was previously adopted for the Avrami’s law [13, 14, 16, 17], and it is calculated according to (3):
$$ R_{\% } = \sqrt {\frac{{\sum\limits_{t = i} {\left( {X(t = i) - X_{{\exp_{m,i} }} } \right)^{2} } }}{{\sum\limits_{t = i} {X_{{\exp_{m,i} }} } }}} \cdot 100 $$
(3)
where X(t = i) and \(X_{{\exp_{m,i} }}\) represent the calculated and the average experimental data at time t = i, respectively. This value expresses the deviation between experimental data and simulated one, that is, the more it tends to zero the more the analytical model overlaps the measured data.
For the second step, the percentage of the values resulting out of the experimental range is run according to (4):
$$ \left( {\frac{{{\text{number of }}X_{{{\text{out}}}} }}{{{\text{number of }}X_{{{\text{tot}}}} }}} \right)_{{G_{{\text{p}}} }} $$
(4)
comparing the number of times values are out (Xout) to the number of total values (Xtot) resulting in each specific growth phase Gp (i.e. latency, exponential and stagnation).
To avoid subjective interpretations, the discretization of the average experimental data into the three phases is run according these steps:
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1.
the total slope of the experimental data mtot was determined as the linear incremental ratio between the starting point (0;0) and the ending point (tend; Xmax) from the experimental data, where tend corresponds to the last measuring time.
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2.
the ith slope mi is determined between the covered area at time i and the previous measure at time i − 1;
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3.
the three phases are evaluated according to condition (5)
$$ \left\{ {\begin{array}{ll} {{\text{Exponential}}} & {m_{i} > m_{{{\text{tot}}}} } \\ {{\text{Latency}},{\text{ Stagnation}}} & {m_{i} \le m_{{{\text{tot}}}} } \\ \end{array} } \right. $$
(5)
This discretization method defines the exponential phase as the phase in which the growth slope mi is higher than the overall linear growth (mtot). Conversely, both the latency and stagnation take place when mi is equal or lower than mtot, respectively, right before and after the exponential phase. Figure 1 shows an example of such discretization: the first 11 experimental data and the last 9 values are grouped respectively in the latency/stagnation phase, since their mi values are always lower than mtot; conversely, the remaining experimental values can be grouped in the exponential phase because their mi are higher than the mtot. In this way, it is possible to define a latency phase where the coverage is not a constant equal to zero, but it has an incremental ratio, even if small.
The goal of the last comparison is to evaluate eventual trend of under/over estimation for such out values and, thus, to asses if one of the models is closer to the experimental data, even when not properly overlapping the data. For every ith out values, the underestimation/overestimation is calculated by determining the difference between the calculated X(t = i) and the minimum/maximum experimental value among the three sample \((X_{{\exp_{1} }} ,...,X_{{\exp_{3} }} )_{i}\) according to (6):
$$ \left\{ {\begin{array}{*{20}c} {\frac{{X(t = i) - \min (X_{{\exp_{1} }} ,...,X_{{\exp_{3} }} )_{i} }}{{A_{{\text{c}}} /A_{{\text{t}}} }}, \, \quad {\text{if}}\, \, X(t = i) < \min (X_{{\exp_{1} }} ,...,X_{{\exp_{3} }} )_{i} } \\ {\frac{{X(t = i) - \max (X_{{\exp_{1} }} ,...,X_{{\exp_{3} }} )_{i} }}{{A_{{\text{c}}} /A_{{\text{t}}} }},\quad {\text{ if}}\, \, X(t = i) > \max (X_{{\exp_{1} }} ,...,X_{{\exp_{3} }} )_{i} } \\ \end{array} } \right. $$
(6)
Moreover, a normalization of such differences to the total covered area Ac/At is set in order to have comparable results. In fact, the total covered area significantly differs among all the materials, ranging between 0.10 and 1.00 [13,14,15,16]. Condition (6) is determined for both the models. Boxplot analysis is run to describe the trend and distribution of such values for each phase.
Overcoming the Avrami’s flaws
The first step of this section wants to validate the hypothesis that the Avrami’s model is not able to correctly simulate microalgae growth for ANt, ACu and AAg materials because the latency phase is missing [15]. In order to verify that, the discretization above is applied to such materials by verifying the presence/absence of the latency phase. Subsequently, by determining the logistic curves for such materials, the work verifies whether the logistic model is able to overcome this flaw. A graphical test is also adopted to check the overcoming of the second Avrami’s flaw for all the materials with the latency time higher than 0, for both conditions 1 and 2 reported in Sect. 2.1.
Correlation with the influencing factors
The third comparison is run to assess which model is lesser influenced by the microalgae influencing factors such as porosity and roughness, surface treatments, as well as different environmental conditions (temperature). To evaluate the correlation with each factor alone, three subsets are formed:
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1.
Porosity and Roughness subset: with all the untreated material under T = 27.5 °C;
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2.
Temperature subset: all the untreated material under T = 10 °C and T = 27.5 °C, respectively;
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3.
Surface Treatment subset: all the treated and respectively untreated materials under T = 27.5 °C.
Three categories are correlated to each subset. The first one is the numbers of values inside the experimental range, the second one involves the fitting quality index R% (–) and the third one considers the values out according to each growth phase (latency, exponential and stagnation phase).
In particular, the effect of porosity and roughness is considered as a combined effect through a fitting surface determined as a 1st degree polynomial equation fitted by using MATLAB R2017b software [31]. A linear regression is considered for temperature and surface treatments. Since this last one is a binary regressor (untreated/ treated), binary indicator variables are used respectively 0 for the untreated materials and 1 for the treated ones [32]. The coefficient of determination R2 (–) is used to assess if a correlation between each model and the cited above influencing factors is present (R2 ≥ 0.50) [32] and the relative trends are then evaluated through scatter plot, only in affirmative cases.