Analysis of heat transfer and AuNPs-mediated photo-thermal inactivation of E. coli at varying laser powers using single-phase CFD modeling Open Access

A

= area of irradiation (m²);

$Cabsi$

= absorption cross-section (m²);

(Cp)_bac

= heat capacity of bacteria (J·kg⁻¹·K⁻¹);

Cp_eff

= effective heat capacity (J·kg⁻¹·K⁻¹);

Cp_w

= heat capacity of water (J·kg⁻¹·K⁻¹);

$g→$

= gravitational force (m·s⁻²);

h_ext

= external heat transfer coefficient (W·m⁻²·K⁻¹);

$I↔$

= unit tensor;

I_o

= initial intensity of laser (W·m⁻²);

k_bac

= thermal conductivity of bacteria (W·m⁻¹·K⁻¹);

k_eff

= effective thermal conductivity (W·m⁻¹·K⁻¹);

k_w

= thermal conductivity of water (W·m⁻¹·K⁻¹);

$Lp−hi$

= interaction length of the i-particle (m) (here: 23 nm);

m_Au

= refractive index of gold (-);

m_gl

= refractive index of glass (-);

n₀

= bacterial concentration (CFU/mL);

N₀

= initial number of bacteria (-);

N_inac

= number of inactivated bacterial cells (-);

N_surv

= number of surviving bacterial cells (-);

p

= pressure (Pa);

$q→$

= heat flux (W·m⁻²);

$q˙$

= external heat source (W·m⁻³);

S

= shape factor in thermal conductivity equation (-);

T

= temperature (K);

T_f

= temperature of fluid (K);

t

= time (s);

V_bac

= bacterium volume (m³);

V_cell

= volume of cells reaching or exceeding 55.0°C (m³); and

$v→$

= velocity vector (m·s⁻¹).

$αeffi,j,L$

= effective polarizability of the i-particle;

$αi,j,L$

= polarizability of the i-particle;

β

= volume expansion coefficient (K⁻¹);

ϵ

= emissivity (-);

θ

= incident angle, here: 0 rad;

μ

= viscosity (Pa·s);

μ_bac

= viscosity of bacteria (Pa·s);

μ_eff

= effective viscosity (Pa·s);

μ_w

= viscosity of water (Pa·s);

ξ

= AuNPs concentration in the considered volume (m⁻³) (here: ξ = 3.0 · 10²² m⁻³);

ρ

= density (kg·m⁻³);

σ

= Stefan–Boltzmann constant (W·m⁻²·K⁻⁴);

σ_abs

= absorption coefficient in room temperature (m⁻¹);

ϕ

= volume fraction of bacteria (-); and

ϕ_m

= maximum packing fraction of bacteria (-).

o

= at initial and standard conditions (24°C, 101 325 Pa);

f

= fluid;

i

= i-particle, in reference to size distribution;

m

= maximum;

w

= water;

abs

= absorption;

bac

= bacteria;

cell

= cell – in reference to bacteria;

eff

= effective;

ext

= external;

inac

= inactivated, in reference to bacteria; and

surv

= surviving, in reference to bacteria.

Au

= gold;

AuNPs

= gold nanoparticles;

AuNRs

= gold nanorods;

CFD

= computational fluid dynamics;

NPs

= nanoparticles;

UDF

= user-defined function; and

UDS

= user-defined scalar.

Considerable research investigating bacterial inactivation has been carried out due to the outbreak of a new coronavirus disease, which may play a very important role in the prevention and management of bacterial infections and can ultimately contribute to the promotion of public health. Various effective methods of bacterial inactivation, including temperature treatments, chemical disinfection, UV radiation, filtration and lasers, have been developed (Ma et al., 2023). The use of laser technology to generate high temperatures or reactive oxygen species is sufficient to eliminate microorganisms. This method of sterilization is accurate and effective. In recent decades, lasers have been used in different ways, and today, different types of lasers, including those combined with gold nanoparticles (AuNPs), are used for bacterial inactivation and virus reduction. This technique is based on the unique attributes of AuNPs, which use light energy to generate heat, and this heat is used to eliminate microbes locally (see works by Zaccagnini et al., 2023; Petronella et al., 2022; and Chavan et al., 2023).

Although experimental methods have made a great contribution to this domain (Van Asselt and Zwietering, 2006; McGuigan and Conroy, 1998; Laroussi and Leipold, 2004), there are limits to a comprehensive understanding of the thermal dynamics involved in bacterial inactivation. Most research in this field relies heavily on experimental techniques. Hence, there is a need for computational fluid dynamics (CFD) techniques to improve our understanding of these processes (Karthikeyan et al., 2021).

This paper discusses the practical use of CFD to simulate the thermal inactivation of bacteria, using a continuous approach supported by proven success. Before delving into the details of CFD applications, it is important to first explore the broader context of studies into the thermal inactivation of bacteria. This inactivation refers to high temperatures that trigger protein denaturation, membrane damage, nucleic acid disruption, loss of cellular fluids and disruption of metabolic pathways. All these processes lead to the inactivation of bacterial cells (Smelt and Brul, 2014). A study on the use of rainwater in hot water systems showed that 55–65°C effectively eliminates harmful bacteria (Spinks et al., 2006), emphasizing a minimum temperature of 60°C for system operation. By testing a thermal and microbial model, a study conducted by Mackey et al. (2006) showed that physical factors such as temperature and cylinder diameter have a greater impact on bacterial removal than biological variables. An article published by Fernandez et al. (2001) examined Bacillus cereus spores and revealed that faster heating rates increased resistance. Nonisothermal activation at 2 K/min resulted in higher D values than isothermal activation, impacting germination and resistance.

In the CFD simulations of bacterial inactivation, the continuum approach is a fundamental methodology. This technique involves discretizing the fluid into a control volume of selected size, generally a few micrometers in length (the fundamental unit of calculation). Each control volume contains a large number of individual atoms. Using the continuum approach means that the fluid can be treated as a continuous substance (Rapp, 2022). This methodology has been applied in both macro- and microfluidic simulations (examples of works in this context can be found in Bayraktar and Srikanth, 2006; Badur et al., 2018 and Ziółkowski and Badur, 2018a, 2018b), and it gives high-quality results. However, to accurately capture microscale phenomena occurring in microfluidic flow, some integrated models can be applied, e.g.: – the slip-velocity model (this model has been well described in articles by Badur et al., 2019, and Ziółkowski and Badur, 2014), – and angular velocity model (Badur et al., 2015). At extremely small length scales, however, this remains less accurate, requiring more detailed approaches such as molecular dynamics simulations. Despite this, for practical microfluidic applications, the continuum approach remains a robust method for studying heat transfer (Lockerby et al., 2015).

The use of CFD techniques in the field of bacterial inactivation can be implemented using two distinct approaches to define the bacteria within the fluid domain. Both configuration approaches are described in Zhao et al. (2008), Ducoste et al. (2005) and Memarzadeh et al. (2004):

• Lagrangian approach: This approach considers bacteria as a discrete phase (Yang et al., 2021; Sozzi and Taghipour, 2006; Ducoste et al., 2005), which allows tracking their individual trajectories through the fluid. It is more commonly applied in scenarios where the behavior of each bacterial cell is needed, such as interaction with surfaces or biofilm growth.

• Eulerian approach: In this approach, the bacteria are treated as an entities of the fluid phase (see Feurhuber et al., 2019; Feurhuber et al., 2021; Feurhuber et al., 2017). This approach is computationally less expensive and is more accurate for scenarios where the individual behavior of bacteria is less critical.

It should be noted that the Eulerian approach has been extensively used in numerous studies. In the context of CFD simulation of bacterial inactivation using Ansys Fluent software (ANSYS Inc., 2021), in addition to the standard equations of fluid dynamics and heat transfer, the Eulerian method requires adding another governing equation, such as a user-defined scalar (UDS) (see Feurhuber et al., 2019; Feurhuber et al., 2021; Feurhuber et al., 2017). Furthermore, user-defined functions (UDFs) to describe bacterial inactivation as a result of temperature increase or UV exposure must be added. As already indicated, the Eulerian approach is more advantageous when the objective is to illustrate bacterial reduction due to temperature increases or other factors.

The CFD simulation of scenarios where two or more materials are contained in the same physical domain typically involves using one of the known approaches, i.e. the single-phase and two-phase approaches (Liang and Mudawar, 2019). However, despite the accuracy achieved with the two-phase approach, the single-phase approach is widely used due to its simplicity of implementation and satisfactory results (Jalali and Karimipour, 2019; Koulali et al., 2021; Yashkun et al., 2021). Based on the literature review, it was found that the single-phase approach is mainly used when the behavior of the whole system is the objective. In connection with the present work, treating bacteria and water as one single medium simplifies the modeling and reduces computational costs.

In the exploration of microchannel and microchamber heat transfer, the continuum approach has been used in works by Abdollahi et al. (2018); Chai et al. (2013); Chai and Wang (2018); Paramanandam et al. (2024); and Zhang and Liu (2021). In these studies, fluids are confined in microenvironments, and various effects were taken into account, such as entrance effects, conjugate heat transfer, viscous heating and temperature-dependent thermophysical properties. Another example of using the Eulerian approach in microfluidic simulation is the work by Abdollahi et al. (2018), where a 3D CFD simulation based on the finite volume method was used to simulate the laminar flow and heat transfer characteristics in interrupted microchannel heat sinks.

In addition, the continuum approach facilitated a detailed examination of optimal rib dimensions and positions, highlighting the importance of geometry and layout variables (Chai et al., 2013). A numerical simulation of nanofluid flow emphasized the influence of rib shape and nanoparticle volume fraction, demonstrating the need for a nanoparticle volume fraction of at least 2% to improve heat transfer (Abdollahi et al., 2018). The thermal-hydraulic performance of finned microchannel heat sinks was assessed in detail, revealing a significant reduction in total thermal resistance (4–31%) and entropy generation rates (4–26%) in finned, interrupted microchannel heat sinks in transverse microchambers, compared with straight microchannels (Chai and Wang, 2018).

Another aspect to consider is that gold nanorods (AuNRs) have been extensively studied for their ability to efficiently convert light into heat [see: (Akbadi et al., 2021; Radomski et al., 2023; Tahghighi et al., 2020) (Zhang et al. (2023))], particularly when their dimensions are optimized to match specific wavelengths in the near-infrared region (NIR). Jain et al. (2006) showed that AuNRs with elongated shapes exhibit strong tunable optical resonance in the NIR, significantly enhancing their light absorption and heat generation properties. This tunability is primarily influenced by the nanorods’ aspect ratio (AR), with higher ARs leading to more efficient photothermal conversion. Huang et al. (2009) further emphasized the role of aspect ratio in determining the localized surface plasmon resonance (LSPR) of AuNRs. Their findings indicated that AuNRs with an AR around 3.4 are particularly effective at shifting the LSPR into the NIR, where wavelengths such as 808 nm are found. This ability to shift the resonance by adjusting the length-to-diameter ratio allows for the fine-tuning of AuNRs for maximum heat generation under NIR irradiation. Due to this fact this wavelength was chosen for analyzing in this article.

Further supporting these findings, Baffou and Quidant (2013) reviewed the principles of thermo-plasmonics, explaining how AuNPs act as efficient nanoscale heat sources when illuminated at their LSPR. Guglielmelli et al. (2021) added insights on the role of thermo-plasmonic AuNPs in modern optics and biomedicine, highlighting their potential in advanced applications. Moreover, Zhang et al. (2023) provided a quantitative analysis of the shape effects on the thermo-plasmonic properties of gold nanostructures, showing that nanorods with higher ARs exhibit superior light-to-heat conversion. Diallo et al. (2020) emphasized the influence of size and shape on photothermal heat generation, confirming that AuNRs exhibit excellent photothermal performance in the NIR. In addition, it can be noted that not only AuNRs are suitable for use to high energy conversion but also other types of nanocomposites (Petronella et al., 2016) or surfaces using liquid crystals (Petronella et al., 2023).

In our experiment, we selected AuNRs with dimensions of 15 nm in diameter and 55 nm in length, giving an AR of 3.67. This AR was deliberately chosen to ensure optimal absorption in the NIR, specifically at 808 nm, a common wavelength used in photothermal applications.

In our study, we explore a new area of CFD simulation related to bacterial inactivation. The motivation for this work stems from the need for more effective methods in the field of bacterial inactivation, especially after the spread of COVID-19. The objective of this work is, first, to determine how AuNPs can efficiently convert laser light into heat. The resulting heat source can be used for various applications, including bacterial inactivation in microfluidic devices, which is the focus of this research. A mathematical model was integrated into the CFD simulations to describe the conversion of light into heat and its effect on bacterial dynamics. Furthermore, to account for the presence of bacteria in the water, the single-phase approach was adopted. Our research is structured as shown in the flowchart in Figure 1. The geometry used originated from an experimental laboratory, and the boundary conditions were determined based on real-world scenarios. The light-to-heat conversion model integrated into the CFD simulation using appropriate UDFs was validated against experimental results. The single-phase approach to the bacteria-water mixture was applied using UDFs integrated into the CFD simulation, along with the bacterial inactivation model.

For bacterial inactivation, the microchamber device presented in this study combines the advantages of microfluidics and a targeted treatment strategy. It is constructed by combining a 1-mm-thick borosilicate glass plate (75 × 25 mm²) with a corresponding 1-cm-thick PolyDiMethylSiloxane (PDMS) layer [see: (Petronella et al., 2022) and (Sforza et al. (2024))]. AuNRs are placed at the base of the chamber, treated as a thin film 23 nm thick. By exploring two microchamber heights (10 μm and 100 μm), the physical model can be used to study the germicidal properties of the system.

Figure 2 outlines the chamber’s dimensions, highlighting its structural properties. The laser beam directed onto the device, as illustrated in Figure 2a, uses a targeted irradiation approach, focusing specifically on a 3 × 3 mm² surface of the AuNPs plate (Figure 2b).

We used data from the work of Petronella et al. (2022) to determine the concentration of gold nanoparticles. In the work of Petronella et al. (2022), the gold nanoparticles were not only analyzed in detail for their distribution and morphology, but were also used in a functional system to inactivate bacteria and to detect their presence even at low concentrations. Thus, based on the work of Petronella et al. (2022), we are able to conclude that nanoparticles, when used appropriately, show multifunctionality for both detection of bacteria and their subsequent inactivation. The experimental data allowed the determination of the concentration of ξ = 1.41·10²¹m⁻³. This parameter was then introduced into the mathematical model obtaining the same number of AuNRs. However, it should be mentioned that this assumption introduces a certain simplification, as in the mathematical model the distribution of nanoparticles is uniform. Furthermore, implemented model does not take into account the absorbance shift due to the presence of E. coli bacteria, as illustrated in a recent paper by Sforza et al. (2024). However indeed, the microchamber device is a hybrid solution that effectively integrates the advantages of microfluidics and targeted treatments. Carefully dimensioned, the device enables control and efficiency of bacterial inactivation.

However, it should be noted that microfluidic devices, often referred to as lab-on-a-chip systems, have reshaped experimental models, particularly in the field of biological and chemical analysis (Wang et al., 2015). These miniaturized platforms offer a wide range of advantages, including precise control of fluid dynamics, reduced sample volumes and accelerated reaction times (Hajmohammadi et al., 2018; Li et al., 2019; Feng et al., 2020). In the context of bacterial inactivation studies, these qualities become essential, as they help to improve heat transfer control and overall experiment efficiency (see Patinglag et al., 2021; Pudasaini et al., 2021; Wang et al., 2020). The reduced sample volume in microfluidic devices allows rapid temperature changes, which are essential for obtaining precise, controlled heat transfer conditions. The increased surface-to-volume ratio and confinement of microfluidic flow channels optimize heat exchange, ensuring targeted heat treatment with minimal energy dispersion. Operating under laminar flow conditions, these devices offer predictable and stable flow patterns, facilitating controlled heat transfer processes.

In addition, germicidal chambers represent a distinct category of disinfection systems designed to effectively inactivate microorganisms, including bacteria. These chambers exploit various technologies, such as ultraviolet (UV) irradiation or chemical agents, to achieve germicidal effects (Stephan, 2017; Xu et al., 2013). The controlled environment of germicidal chambers ensures constant exposure to disinfecting agents, maximizing their effectiveness (Won et al., 2023). Although these chambers excel at large-scale disinfection, they can encounter difficulties in reaching complex structures or delivering targeted treatments.

In this work, the specific heat of PDMS used to build the microfluidic chamber is depended on the temperature variation (Narottam, 1986; Agari et al., 1997; Cahill et al., 1991; James, 1999). All thermal properties of the used materials are summarized in Table 1.

Additionally, the thermophysical properties of water and E. coli bacteria are presented in Table 2.

In the present research three density models were used to define the water density:

• The first is the constant density model (ρ = 998.2).

• The second model uses the Boussinesq equation, expressed as follows (Braginsky and Roberts, 2007; Barletta, 2022):

  $ρ(T)=ρ0(1−β(T−T0))$

  (1)

where:

ρ= density as a function of temperature T;

ρ₀ = reference density at a reference temperature T₀;

β = coefficient of thermal expansion; and

T = current temperature.

Therefore, using the Boussinesq equation, the density is considered constant except in buoyancy forces, where the density of water varies following the Boussinesq approximation:

• The third density model considers the density temperature-dependency (Wagner and Pruss, 2002), described as follows:

  $ρ(T)=999.9720+0.3230⋅T−0.0013·T2+0.00000175⋅T3$

  (2)

The specific thermo-physical properties of E. coli bacteria come from literature sources, as follows: density (Martinez-Salas, Martin, and Vicente, 1981), specific heat capacity (Lee and Kaletunc, 2002), thermal conductivity (Nakanishi et al., 2017), respectively.

The single-phase approach simplifies the modeling of the bacteria-water mixture by treating it as a homogeneous medium, which significantly reduces computational complexity and cost. This approach is justified because the focus is on the overall behavior of the system, rather than the detailed interaction between individual particles and the fluid. The use of the effective thermal properties allows the contribution of bacterial characteristics to be taken into account in the overall diffusion of heat inside the microdevice (Buongiorno, 2006; Feurhuber et al., 2019). The flow of the bacteria-water mixture inside the microfluidic device is characterized by a 3D, laminar regime in a transient state. The continuum equation for this system encompasses (Darrigol, 2002; Brenner, 2005):

where:

ρ_eff = effective density, kg·m⁻³;

t = time, s; and

$v→$ = velocity vector, m·s⁻¹.

This equation ensures mass conservation, depicting the temporal change of effective density (ρ_eff) and divergence of the product of density and velocity. The effective density of the mixture (bacteria-water) depends on the volume fraction ϕ of bacteria, and it is obtained using different empirical correlations (in this case, the rule of mixtures to calculate effective density):

where:

ϕ = volume fraction of bacteria;

ρ_w = density of water, kg·m⁻³; and

ρ_bac = density of bacteria, kg·m⁻³.

The momentum equation for viscous fluid (Reynolds, 1883), thence, is written as follows (Buongiorno, 2006; Feurhuber et al., 2019):

where:

p = pressure, Pa;

μ_eff = effective viscosity, Pa·s;

$I↔$ = tensor unit;

$g→$ = gravitational force, m·s⁻²; and

⊗ = dyadic multiplication.

Describing momentum conservation, it accounts for the effects of pressure, viscosity and gravity on the flow. Moreover, the effective dynamic viscosity of the mixture was estimated using the Brinkman model (Brinkman, 1952):

where:

μ_w – viscosity of water, Pa·s.

Likewise, the energy equation can be presented by Buongiorno (2006):

where:

Cp_eff = effective specific heat capacity, J·kg⁻¹·K⁻¹;

T_f = temperature of fluid, K;

k_eff = effective thermal conductivity, W·m⁻¹·K⁻¹; and

$q˙$ = external heat source, W·m⁻³. The source of energy can represent absorbed energy from a laser.

In this work, the source term on the boundary by the UDF used the DEFINE_PROFILE macro has been applied. Consequently, the source term in the energy equation can be eliminated. It is well known that the energy conservation law states that the total energy of the system equals the sum of work and heat added to the system. However, since the water is working fluid and it can be considered as low-viscosity flows (viscous dissipations typically minimal), also due to time-efficient calculation and physical characteristics of the system in Figure 2, the viscous dissipation and pressure work terms in the energy equation for water flow were neglected. Therefore, the applied equation can be simplified to the form:

Here, the rule of mixtures to calculate the effective heat capacity is according to the equation:

where:

Cp_w = specific heat capacity of water, J·kg⁻¹·K⁻¹; and

Cp_bac = specific heat capacity of bacteria J·kg⁻¹·K⁻¹.

To predict the effective thermal conductivity of the water-bacteria mixture, the Lewis-and-Nielsen model (Pal, 2008) can be used as follows:

where:

S = bacteria shape factor;

ϕ_m = maximum packing fraction, depending of alignment’s type;

k_bac = thermal conductivity of bacteria, W·m⁻¹·K⁻¹; and

k_w = thermal conductivity of water, W·m⁻¹·K⁻¹.

S and ϕ_m can be obtained by referring to the following Table 3 and Table 4.

As described in Section 2, “Analyzed germicidal microchamber system,” and Figure 2, the slab film of AuNPs deposited at the device bottom was irradiated by a laser of wavelength λ = 808 nm, with an intensity I₀ corresponding to a power P. The conversion of electromagnetic radiation to thermal energy in a medium containing AuNRs can be described by considering the absorption characteristics of the particles (Radomski et al., 2021; Radomski et al., 2024). The total energy generated by the whole system was determined using discrete particle integration, which allows us to account for all AuNPs present in the domain. Here, the general equation for heat generation is given as:

Where:

$σabsi$= absorption coefficient of the i-th AuNPs (m⁻¹);

M = number of discrete absorbing element AuNPs (-); and

$Iabsi$= intensity of light absorbed by i-th AuNPs (W · m⁻²).

Discrete particle integration reflects the accumulated effect of individual particles. It should be added that the bottom layer with AuNRs was functionalized according to the protocol in Zaccagnini et al. (2023). The functionalization provides an electrical interaction with a strong bonding that prevents floating around the chamber freely. Likewise, the distance between each nanoparticle is circa 70 nm, which is greater than their effective size, and which brings the information that the mutual interaction between nanoparticles is significantly weaker than the functionalized surface, and which prevents the tendency to a nanoparticles’ agglomeration.

The absorption coefficient of the i-th AuNPs can be determined using scattering and extinction cross-section of the concerned particle:

We use the Rayleigh–Drude approximation to determine both scattering and extinction coefficients:

where:

λ = wavelength (nm);

ξ = AuNRs concentration in the considered volume (m⁻³);

$Cexti,j,L$ = extinction cross section of the i-particle, m²;

$Cscai,j,L$ = scattered cross section of the i-particle, m²; and

$αeffi,j,L$ = effective polarizability of the i-particle.

$αi,j,L,i^$ is the polarizability of the i-th AuNRs (m³) for the considered axis $i^$⁠, which is determined using the following formula (Radomski et al., 2024):

where:

$Vci,j$ and $Vgi,j$–Volume of the i-th AuNPs and capping agent (m³);

ε_c–the permittivity of the gold - core (-);

ε_h–the permittivity of the host medium (-);

ε_g–the permittivity of the agent medium (-); and + indicates the oriented direction amongst the Cartesian coordinates, x, y, z. Depolarization factor (Л_i,j) might indicate that the opposite charges prefer to be accumulated at edges, generating dipoles. The effective polarizability is defined as follows (Radomski et al., 2024):

The intensity of the light absorbed by the i-th AuNP depends on the initial intensity of the incident laser light and the optical properties of the medium as well as the distance traveled. As the Lambert–Beer–Bourger law states, laser intensity decreases exponentially with distance traveled through an absorbing medium, the intensity of light absorbed by the i-th AuNP can be determined using of the following formula:

where:

I₀ = initial intensity of laser (W · m⁻²);

R₀ = reflection coefficient of glass (-); and

R_Au = reflection coefficient of AuNRs (-).

The term $(1−sin2θ⋅(mAu2mgl2))1/2$ in the equation (17) represents the effect incident angle, where: m_Au and m_gl are refractive indices of gold and glass, respectively. The model presented in equations (12) – (17) was verified on a different geometry, under different conditions according to data from publications: Radomski et al. (2024) and Zaccagnini et al. (2023). Such satisfactory agreement was obtained for the AuNRs absorption spectrum and the temperature field on the studied geometry that it was decided to use the model mentioned in this work as well.

Based on equations (11) – (17), the heat generation rate due to a laser irradiation in AuNPs was calculated using the following equation:

The thermal boundary condition at this area can be described as heat flux:

However, for initial conditions, it is completed by the condition at all contact surface, which should combine both conditions to have the heat flux continuity:

For the glass bottom in Figure 2, the mixed convection/radiation condition can be implemented: [see (ANSYS Inc., 2021)]:

where:

σ = Stefan-Boltzmann constant, (W · m⁻²K⁻⁴);

ϵ = relative emissivity (-); and

h_ext = external heat transfer coefficient (W · m⁻²K⁻¹).

On the other hand, for fluid-solid interfaces, in the simplest form, the heat flux continuity is described by the equation:

It should be added that in the presented simulations, at the interfaces between fluid and solid, as well as between different solid phases, the continuity of heat flux is maintained. This is expressed mathematically through the equality of the product of thermal conductivity (k) and temperature gradient (∇T) on both sides of the interface. Specifically, for the interface between fluid (f) and the i-th solid phase (S_i),

However, additional phenomena can occur, like temperature jump (Smoluchowski, 1898), or phase transition (Błauciak et al., 2021; Ziółkowski, 2019), especially in porous structures. Similarly, within the i-th solid phase, the continuity of heat flux is expressed as:

These conditions ensure that the heat flux remains constant across the interfaces, allowing for a smooth exchange of thermal energy between different regions of the system. For other walls included in Figure 2, the following boundary condition is implemented:

This equation set has been extended to represent bacterial inactivation. Numerical procedures to implement it are presented in the next section.

In our simulations, a no-slip boundary condition was applied at all walls of the microchamber. This condition assumes that the fluid in direct contact with the chamber walls has zero velocity relative to the walls, meaning that there is no relative motion between the fluid and the solid boundaries:

Detailed parameters related to the boundary conditions together with the initial conditions are provided in Table 5.

Numerical simulations were carried out using the commercial software Ansys Fluent (ANSYS Inc., 2021). For 3D flow within the microchamber, a transient flow regime coupled with a laminar flow condition was assumed. The transient regime was also adopted for conduction across the solid parts. The SIMPLE algorithm was used for velocity-pressure coupling in the Navier–Stokes equations. The Second-Order Upwind algorithm was used for the spatial discretization of pressure, velocity and energy terms. In addition, the transient formulation was carried out using the Bounded Second-Order Implicit algorithm. To ensure the independence of results from the time step size, four simulations were run with different time steps, using the same boundary conditions. The average temperature results at t = 190 s inside the microfluidic device for each time step size are shown in Table 6. The table shows that the time step Δt = 0.0008 presents a good compromise between result accuracy and computational costs. Therefore, a time step of Δt = 0.0008 was used for the rest of the simulations carried out in the present work.

A structured mesh grid was used in the simulation of the microdevice to enhance computational efficiency. Special attention was given to mesh treatment at the microchamber boundaries due to the substantial temperature gradient in these regions (refer to Figure 3). The bias technique was applied, and the bias factor was computed to ensure that the size of the first element near the microchamber wall was equaled the dimensions of the elements in the microfluidic device. To assess the sensitivity of numerical results concerning velocity within the chamber to the node number, various grid systems were tested. Figure 4 illustrates the results in terms of the maximum fluid velocity at different simulation times for different mesh grids. Notably, the maximum velocity of the bacteria-water mixture exhibited insensitivity to the mesh grid size beyond the M4 grid. This insensitivity indicates that the simulation results have reached a level of mesh independence, meaning that further refinement of the mesh does not lead to significant changes in the results. Consequently, the M4 mesh grid was used for the subsequent numerical simulations. This choice of the M4 grid strikes a balance between computational efficiency and accuracy. By selecting a mesh that is fine enough to capture all relevant flow details but not so fine as to unnecessarily increase computational cost, we ensure that the simulations are both reliable and efficient.

To take into account, the thermophysical properties of E. coli bacteria in the microchamber, UDFs have been developed using the equations defined in equations (4), (6) and (10) with the DEFINE_PROPERTY macro, and another UDF with the DFINE_SPECIFIC_HEAT macro to define the heat capacity of the bacteria-water mixture [equation (9)]. It should be noted that these UDFs also take into account temperature dependency.

In addition, a UDF with the DEFINE_PROFILE macro has been developed to integrate the boundary condition defined in equation (19), using the heat generation formula [equation (18)]. This UDF calculates the heat generation rate of the AuNRs deposited on the bottom surface of the microchamber, using the assigned input data. The result will be used as heat generation at individual mesh elements. The UDF also contains a conditional loop to select the mesh cells responsible for heat generation, given that the irradiated surface is 3 × 3 mm².

The volume of cells exceeding 55°C was tracked to calculate inactivated bacteria in the microfluidic chamber using the equation:

where:

V_cell = volume of cells reaching or exceeding 55°C; and

V_bac = volume of a single E. coli bacterium.

Meanwhile, the number of surviving bacteria, N_surv, is given by:

where:

N₀ = initial number of bacteria; and

N_inac = number of inactivated bacteria.

An initial bacterial concentration, n₀, is given by:

where V_cell – volume of chamber. It is assumed that bacteria subjected to a temperature greater than or equal to 55°C for a period of one minute will be inactive. To measure the number of bacteria inactivated inside the microchamber during simulation, a UDF was developed and integrated into Ansys-Fluent using the DEFINE_ON_DEMAND macro. The UDF iterates through the fluid domain, measuring the temperature of each mesh cell. If the temperature exceeds the specified threshold, it calculates the contribution of that mesh cell to the total volume of inactive bacteria using equation (27).

In this work, the validation was carried out by comparing the results of the numerical simulations with those obtained experimentally. The experimental procedure carried out in the nanorods experiment consists of applying a layer of AuNRs to the lower surface of the microchamber, as described in Figure 2. The nanoparticle layer is then irradiated by a laser of power P = 0.8W with a wavelength of 808 nm (see Table 6), for 190 s. A thermal camera was used to record the maximum temperature at the bottom surface of the borosilicate glass and the corresponding temperature fields. The results obtained from the CFD were compared with those obtained using a thermal camera, as shown in Figure 5. The thermal fields obtained numerically agree very well with those obtained experimentally. In a further comparison, the evolution of maximum temperature during the irradiation process (see Figure 6), shows good agreement between the two time-dependent temperature curves. Both of these comparisons demonstrate the accuracy of the numerical simulations carried out in this work, and enhance the validity of the proposed CFD model.

In the current work, heat transfer within a microfluidic chamber designed to understand the influence of laser-activated AuNRs on the thermal inactivation of E. coli bacteria has been numerically investigated. Three key parameters were taken into account: the initial concentration of E. coli bacteria (see Table 7) characterized by three presence rate values ϕ (0, 0.3 and 0.8), the design of the chamber, with two thickness variations (h = 10 μm and h = 100 μm), and also the laser power (0.2 W ≤ P ≤ 1.0 W).

Within the microfluidic chamber under laser irradiation, this study of heat transfer dynamics shows clear predominance of conduction as the main mechanism. This observation is supported by the isothermal structure depicted in Figure 7 where, after 190 s of irradiation, the isotherms show a tight, parallel configuration. Such a pattern shows the absence of a convective regime, indicating that conduction is the dominant mode of heat diffusion in the confined microchamber. The physical explanation for this phenomenon lies in the constraints imposed by the chamber’s dimensions. With a small height and confined space, the development of natural convection currents is limited or non-existent. Consequently, diffusion conduction appears to be the dominant process. The role of AuNRs is significant in this system, since it considered as the source of heating. Moreover, the laser irradiation at the base of the microchamber, where the AuNPs are deposited, leads to the conversion of electromagnetic energy into thermal energy. This photothermal conversion is highly efficient due to the absorption properties of the AuNRs. The CFD simulations incorporate this by modeling the localized heat generation within the microchamber as a result of the AuNPs’ presence. The discrete particle integration method used in the simulations accounts for the energy contribution of each nanoparticle, allowing for a detailed analysis of the heat distribution. Despite the high localization of heat around the AuNRs, the overall heat transfer within the chamber remains dominated by conduction, as shown by the consistent isothermal profiles. The simulations indicate that, while the AuNPs significantly enhance the localized heating effect, leading to the inactivation of E. coli, this does not alter the macroscopic mode of heat transfer, which is conduction. The analysis of different density models constant density, the Boussinesq approximation and temperature-dependent density further supports this conclusion, showing minimal impact on the mean temperature profiles regardless of the model used. The effectiveness of the AuNRs in achieving photothermal inactivation is thus integrated into the thermal dynamics without shifting the dominant heat transfer mechanism from conduction to convection.

To validate the previous result (heat conduction is the predominant mode), three density models for the bacteria-water mixture were examined, namely, the constant density model, the Boussinesq approximation [equation (1)] and the temperature-dependent density model [equation (2)]. The results in Figure 8 show the mean temperature profile of the bacteria-water mixture vs irradiation time. These results demonstrate that even if we change the density model in the CFD simulations, the mean temperature values remain almost unchanged. Clearly, the choice of density model has a minimal impact on the evolution of mean temperature.

By way of explanation, the constant density model assumes uniform density throughout the mixture without any variation, while the Boussinesq approximation takes into account buoyancy effects due to density variations caused by temperature difference. Even with the temperature-dependent density model, which takes into account the possibility of adapting to varying thermal conditions, the overall temperature behavior remains largely insensitive to these variations. Together, these results emphasize the high dominance of conduction in heat transfer within the microchamber. The confined nature of the experimental device, combined with its specific geometry, underlines the importance of conduction over convection mechanisms.

In this subsection, temperature variations inside the microfluidic chamber under laser irradiation were explored, examining different initial bacterial concentrations (N_o) – see Table 7. Figure 9 shows the time evolution of the maximum temperatures, this can describe the interactions between heat generation, dissipation mechanisms and effects of chamber design. At the beginning of laser irradiation, the heat generation rate exceeds the dissipation, resulting in a rapid increase in temperature. Over time, the mechanisms of dissipation catch up, moving from an initial rapid increase to a more gradual rate. In particular, the chamber height influences the reached temperature, with h = 10 μm exhibiting higher temperatures than with h = 100 μm. Furthermore, it was found that the rate of bacterial presence (volume fraction ϕ) has no significant impact on thermal dynamics within the system. This observation can be attributed to the thermal properties of bacteria, which closely match those of water. However, a subtle effect was noted in the chamber with h = 100 μm, suggesting a limited influence on dissipation dynamics.

In Figures 10–13, the temporal evolution of maximum and mean temperatures in the microscopic chamber reveals a consistent trend as the laser power increases, regardless of the chamber height (h = 10 μm [see: Figures 10 and Figure 12] or h = 100 μm [see: Figure 11 and 13]). The rise in temperature is expected as a consequence of laser irradiation on AuNPs within the microfluidic chamber. Where localized nanoscale heating contributes significantly to the overall temperature increase inside the chamber. The maximum temperature reflects the peak thermal effect reached during the process, whereas the mean temperature provides an average measure. This is consistent with previous experimental Zaccagnini et al. (2023) particularly in the low power range of lasers, where electron saturation has not yet occurred. The observed increase in both maximum and mean temperatures as the laser power rises signifies enhanced thermal inactivation of bacteria, underscoring the pivotal role of AuNRs in facilitating effective photothermal inactivation. This enhancement is particularly pronounced at higher laser powers, where the thermal effects generated by the AuNRs are more substantial, leading to efficient bacterial inactivation.

Figure 14 explores the relationship between laser power and maximum temperature after 190 s of irradiation. Notably, the linear variation suggests a proportional response within the studied power range (0.2 W to 1.0 W), corresponding to power densities of 2.22 Wcm⁻² to 11.11 Wcm⁻². This aligns with experimental findings reported in Zaccagnini et al. (2023), indicating a two-stage behavior in temperature progression. The initial linear phase, consistent with presented results, transitions into a slower increase beyond 12 Wcm⁻².

Since the trends in terms of bacterial inactivation were the same for the 100 μm and 10 μm height microchambers, we chose to present the results for the h = 100 μm height microchamber for more simplicity. Figure 15 shows the changes in the inactivated and surviving bacteria population during laser irradiation. The curves represent the temporal evolution of the above-mentioned variables using different laser powers, thereby providing additional scope for the thermal analysis discussed above. In addition, the spatial distribution of active/inactive bacteria obtained from the calculated total volume gives interesting information on the inactivation rate of bacteria caused by laser irradiation this was presented in Figure 16. It should be recalled that the assumption taken about the condition for which the bacterium will be inactivated states that the bacteria cell will be inactive after exposure to a temperature of T ≥ 55°C for a period of 60 s. Thus, the results presented here represent the beginning of inactivation. The information provided in Figures 15 and 16 shows that at laser powers of 0.8 W and 1.0 W, there is a notable rapid temperature increase in the mesh elements, reaching temperatures of 55°C or higher. Specifically, at a power of 1.0 W, the temperature threshold of 55°C is reached approximately at the 10 s mark, whereas at 0.8 W, this temperature is reached around the 24 s mark. Clearly, as the temperature of the microchamber mesh element reaches or exceeds 55°C, this impacts the bacteria viability. Where, the increase in bacteria inactivation and the decrease in surviving bacteria, as depicted in Figure 15, illustrate the effect of thermal processes induced by laser irradiation on the bacteria inactivation mechanism. This finding confirms the relationship between the thermal impact generated by laser exposure and the process of bacterial inactivation. However, for laser powers ranging from 0.2 W to 0.6 W, the number of inactivated bacteria (Figure 15) as well bacteria inactivation rate (Figure 16) remains null over time, indicating that the entire microfluidic chamber does not reach temperatures above 55°C (see Figure 14). This corresponds to the general trend observed in the temperature analysis, and highlighting the importance of laser power in controlling thermal effects and bacterial inactivation.

The CFD simulation study presented here focused on the AuNRs irradiation technique for the thermal inactivation of E. coli bacteria confined within a microchamber. This problem involves several factors, such as irradiation power, initial bacterial concentration, microchamber design, the transient nature of the process. The light-to-heat conversion model [see: (Radomski et al., 2024)], which is based on three fundamental approximations: discrete particle integration, Rayleigh–Drude scattering and Lambert–Beer–Bourger law, has been validated against experimental results, showing good agreement.

The results provided important observations, namely:

• within the microchamber, the conductive regime dominates heat transfer, according to the three density models used in the CFD simulations;

• the calibration of the model was performed with respect to the AuNRs experiment [see: (Petronella et al., 2022)];

• the role of AuNRs is crucial here, as they effectively convert laser energy into heat, leading to localized temperature increases that are essential for the precise thermal inactivation of E. coli;

• it was found that the maximum temperature reached within the microchamber varies linearly with laser power output, which aligns with previously published experimental results by (Petronella et al., 2022) and (Zaccagnini et al., 2023); and

• obtained results confirms the robustness of the numerical simulations conducted in this study and offers significant perspectives for optimizing bacterial inactivation processes.

Moreover, the detailed exploration of bacterial inactivation dynamics reveals a direct correlation between laser power and the population of inactivated and surviving bacteria. The temporal patterns observed in the bacterial response underscore the importance of laser parameters in achieving effective bacterial inactivation. These findings highlight the pivotal role of AuNRs in enhancing the photothermal effect within the microchamber and provide a solid foundation for further optimization of this technique in microfluidic applications.

This research was partially funded by the National Science Centre in Poland under the project titled “Shape and displacement optimization of gold nanorods in the killing chamber for photothermoablation processes” project number UMO-2021/43/D/ST8/02504. For Open Access purposes, the authors have applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version resulting from this submission. Whole of the theoretical research performed by Gdańsk University of Technology and Medical University of Gdańsk was funded by the National Science Centre, Poland within the scope of the SONATA-17 project No. 2021/43/D/ST8/02504, entitled “Shape and displacement optimization of gold nanorods in the killing chamber for photothermoablation processes”.

Computations were carried out using the computers of Centre of Informatics Computations were carried out using the resources of the Centre of Informatics, Tricity Academic Supercomputer and Network (CI TASK) in Gdansk, Poland.

Piotr Radomski is also grateful to the Doctoral School at Gdańsk University of Technology for providing a scholarship.

Luciano De Sio acknowledges the support of the following: the NATO Science for Peace and Security Programme (SPS-G7425, CLC-BIODETECT); the Air Force Office of Scientific Research, Air Force Materiel Command, U.S. Air Force, under the project “Digital Optical Network Encryption with Liquid-Crystal Grating Metasurface Perfect Absorbers” (FA8655-22-1-7007, P.I. L. De Sio, EOARD 2022–2025); and the Italian PON project TITAN “Nanotechnology for Cancer Immunotherapy” (ARS01-00906, 2021–2023).

Part of the experimental research performed by María C. Nevárez Martínez was funded by the National Science Centre, Poland within the scope of the PRELUDIUM-18 project No. 2019/35/N/ST5/00464, entitled “Novel nanomaterials based on titanium composites conjugated with affibody molecules with potential photothermal conversion application”. This work was performed by María C. Nevárez Martínez, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA, under contract 89233218CNA000001.

Erratum: It has come to the attention of the publisher that the article Koulali, A., Ziółkowski, P., Radomski, P., De Sio, L., Zieliński, J., Nevárez Martínez, M.C. and Mikielewicz, D. (2024), “Analysis of heat transfer and AuNPs–mediated photo–thermal inactivation of E. coli at varying laser powers using single–phase CFD modeling”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/HFF-04-2024-0252, contained errors in the authors names in the copyright section. This error was introduced during the production process. Copyright © 2024, Koulali Aimad, Piotr Radomski, Luciano De Sio, Jacek Zieliński, María Cristina Nevárez Martínez and Dariusz Mikielewicz has been corrected to Copyright © 2024, Aimad Koulali, Paweł Ziółkowski, Piotr Radomski, Luciano De Sio, Jacek Zieliński, María Cristina Nevárez Martínez, Dariusz Mikielewicz. The publisher sincerely apologises for this error and for any confusion caused.