Der Artikel untersucht die kritische Frage des Thermomanagements in dünnen Schichten, die aufgrund der abnehmenden Dicke von Schichten in mikroelektronischen Verpackungen und anderen technischen Anwendungen zunehmend relevant wird. Es untersucht, wie physikalische Parameter wie Wärmediffusivität, Schichtdicke und Konvektionskoeffizienten die Temperaturverteilung unter örtlich begrenztem Wärmefluss beeinflussen. Die Studie verwendet eine ungefähre integrale Formel, die die Funktion der Grünen nutzt, um eine quantitative Analyse und physikalische Diskussion zu liefern. Zu den wichtigsten Ergebnissen gehört die Beobachtung, dass geringere Wärmediffusivität und Schichtdicke zusammen mit größeren Konvektionskoeffizienten zu einer stärker lokalisierten Erwärmung führen. Der Artikel stellt auch numerische Experimente vor, die die theoretischen Ergebnisse bestätigen und die praktischen Implikationen dieser Ergebnisse aufzeigen. Die Forschung unterstreicht die Bedeutung der Optimierung dieser Parameter für ein effektives Wärmemanagement in verschiedenen industriellen Anwendungen und bietet Erkenntnisse, die Innovationen in Materialwissenschaft und Technik vorantreiben können.
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Abstract
In engineering applications, comprehending thermal issues arising from local heat fluxes in thin films has gained significant attention in recent times. For heat transfer processes in applications, it is important to determine physical parameters of the films for which the temperature outside the heating zone does not increase rapidly. For example, microelectronics packages are facing an increased demand for processing large amounts of data, leading to higher energy consumption. Dissimilar materials are used as thin films in these packages, and their thickness is becoming increasingly smaller. These conditions lead to thermal management issues, as local heat flux may occur in the thin films, making heat management more critical. Although some experimental studies have been conducted, in the case of thin films with very small thickness, the difficulty in temperature measurement of thin films increases, resulting in insufficient research. To clarify their physical significance and for studies in situations where measurement is challenging, research through validated mathematical models is necessary. This paper investigates this problem based on an asymptotic analysis for solutions to the heat equation with Robin boundary condition. We show that the temperature outside the heat flux region can be lowered by reducing the film’s thickness, reducing the thermal diffusivity or by increasing the heat convection. The theoretical result is also supported by the numerical experiments. It means that thermal energy may not be released effectively, and the temperature in localized areas, such as heat spots, could rise drastically. Therefore, in engineering applications where the thickness of thin films decreases, thermal management becomes more challenging, and more efforts are needed to increase thermal diffusivity.
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1 Introduction
Understanding thermal behavior caused by local heat fluxes in thin films or plates has gained significant attention. For instance, microelectronics packages face a growing demand to process larger volumes of data, leading to higher energy consumption. These packages often incorporate thin films, of which thickness continues to decrease, reaching very small values. These conditions make thermal management of local heat flux in thin films an increasingly critical issue [1]. Moreover, understanding this thermal behavior is beneficial for a wide range of engineering applications. The recent trend in the metal forming industry is the increased utilization of local heat treatment to enhance the failure limit of materials [2, 3]. In the food industry, infrared (IR) local heat treatment is extensively employed for various purposes [4, 5]. Heat issues in power devices [6], semiconductors [7], and various microelectronics applications [8‐10] are also gaining increasing attention. Consequently, investigating the thermal response and behavior of thin materials under localized surface heat flux is essential for addressing the thermal management in industrial applications. Prior studies on heat flux have explored diverse aspects, such as the measurement of radial thermal conductivity under time-varying heat flux conditions in cylindrical structures [11], the analysis of phase change behavior influenced by transient heat flux using perturbation methods [12], and the impact of time-dependent heat flux on solidification processes considering thermal resistance and convection [13]. Additional research has addressed non-linear heat flux interactions in sublimation models with temperature-dependent internal heat generation [14]. These studies provide valuable insights into heat flux effects relevant to a broad range of engineering applications. This study aims to further investigate the physical parameters of thin films—particularly as their thickness becomes very small—through theoretical analysis based on an analytical solution.
This work examines the influence of physical parameters, including material properties, geometrical characteristics, and boundary conditions, through mathematical analysis of thin films. Methods such as experiments and finite element (FE) analysis are effective tools that have been used in various studies [15‐17], but this study focuses more on analytical solutions for a more theoretical analysis when conducting local surface heat flux on thin films. Based on an approximate integral formula using the Green’s function presented in Lee and Choi [18], this paper provides a quantitative analysis of the local heating and physical discussion. The findings highlight how physical parameters such as thermal diffusivity, film thickness, and convection coefficients impact temperature distribution, underscoring the importance of optimizing these parameters in engineering applications.
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The thin films with surface heat flux can be modeled by the following heat equation with Robin boundary condition:
where \(\mu \) denotes the thermal diffusivity, defined as the ratio of thermal conductivity \(k_\textrm{th}\) to the product of material density \(\rho \) and specific heat capacity \(C_\textrm{p}\) at constant pressure, i.e., \(\mu = k_\textrm{th}/(\rho C_\textrm{p})\). The \(F({\textbf{x}},t): P \times \{0,h\} \times [0,\infty )\rightarrow \mathbb {R}\) is the heat flux and \(a>0\) is the heat convection coefficient divided by the thermal conductivity. The values \(T_0\) and \(T_1\) refer to the temperatures of the outer parts of the upper and lower surfaces of the plate, respectively. In addition, the domain \(\Omega _{h} = P \times [0,h]\) is a film with small thickness \(h>0\) and \(P \subset \mathbb {R}^2\). In (1), no heat transfer is assumed along the side boundary \(\partial P \times [0,h]\) because the surface measure of \(\partial P \times [0,h]\) is much smaller than that of the upper and lower boundary \(P \times \{0,h\}\). We refer to the paper [19] for an existence result of the heat equation with the Robin boundary condition based on the compactness argument.
The extent of local heating is determined by analyzing the temperature difference between the heating zone and the surrounding area. To formulate this problem mathematically, we consider the support of the heat flux F given by
Question A: For each \(x \in A_\textrm{out}\) and \(t>0\), how small is the quantity of the temperature V(x, t) and what is the effect of the physical parameters \(a >0\), \(h>0\) and \(\mu >0\) on the temperature?
The aim of this paper is to answer this question in a rigorous way. To this end, we recall that the solution of (1) can be represented by an integral formula using the Green’s function (see e.g., [20, pp. 596–597] and [21, Appendix B]), which involves a three-dimensional spatial integration of an infinite series sum, depending on the shape of the film P and \(h>0\). Since the formula is not easy to analyze directly, the second and third authors [18] demonstrated that the integral formula can be approximated by a simpler integral formula when \(h>0\) is small by showing that the terms in the infinite series except the first one are negligibly small for small value of \(h>0\).
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Based on the simplified formula of [18], we proceed to address Question A. To do so, it is reasonable to assume that the initial state \(V_0\) of (1) is given by a static state G since we mainly focus on the growth of the temperature derived by the external heat source F. Specifically, we assume that the initial state \(V_0\) is given by a static solution \(G: \Omega _h \rightarrow \mathbb {R}\) satisfying
The existence of such a static solution G can be established by a variational method (refer to [22]), and the time derivative term of G under the static condition becomes zero. Consequently, to answer Question A, it is sufficient to investigate the growth of the difference between the solution V(x, t) and the initial state G:
We will study the effect of a, h, and \(\mu >0\) on the above quantity for \(x \in A_\textrm{out}\) and \(t>0\). Namely, we establish the following theorem in this work.
Theorem 1.1
Suppose that \({\textbf{x}} = (x_1, x_2, x_3) \in A_\textrm{out}\) with \(\text {dist}((x_1,x_2), \text {supp}_y F) = R >0\) and that \(ha \le 1/3\). Let \(M>0\) denote the area of the plate \(M = vol(P)\).
We observe that the integrations in the estimates (6) and (7) are governed by the values of the physical parameters a, h, and \(\mu >0\) as well as the distance \(R>0\) between the point x and the heat source F. The effect of \(\mu >0\) is straightforward in the integrations of (6) as \(\mu \rightarrow 0\). Namely,
which confirms that the heat flux localization becomes better as \(\mu \) is chosen smaller. In contrast, the effects of the parameters a, h are more more complex and less straightforward to characterize. To facilitate a clearer understanding of their effects, we will derive a simpler estimate from the above estimates in the following theorem.
Theorem 1.2
Suppose that \({\textbf{x}} = (x_1, x_2, x_3) \in A_\textrm{out}\) with \(\text {dist}((x_1,x_2), \textrm{supp}_y \;F) = R >0\) and that \(ha \le 1/3\). Then, for any \(t>0\), we have
where we have set \(Q = \int _{\mathbb {R}^2} \textrm{e}^{-\kappa |y|^2} \textrm{d}y\).
We note that the bound (10) is not sharp when the distance \(R>0\) becomes very small, as the right-hand side of (10) diverges to infinity as R approaches zero. It is complemented by the estimate (11) which provides a uniform bound for the heat profile V(x, t) that is independent of \(R>0\).
The value of \(\varvec{\alpha }_1 (h,a)\) is roughly close to \(\sqrt{2a/h}\) if a/h is sufficiently large (see Lemma 2.2). Combining this with the above estimate will allow us to analyze the effect of the physical parameters h and a on the solution to (1) outside the support of the heat source. Precisely, we will confirm that the first term on the right-hand side of (10) goes to zero if we take a limit for one of the parameters as (i) \(h \rightarrow 0\), and (ii) \(a\rightarrow \infty \), while holding other parameters fixed. Overall, we reach the conclusion that the temperature outside the region of heat flux tends to increase more slowly if the thickness of the film h and the thermal diffusivity \(\mu = k_\textrm{th}/(\rho C_\textrm{p})\) are chosen smaller and the heat convection coefficient divided by thermal conductivity a is chosen larger (refer to Sect. 3), giving a concrete answer to Question A. The validity of this conclusion is further supported by numerical experiments.
Our findings are also aligned with physical intuition. A small value of \(\mu \) corresponds to materials with small thermal diffusivity, implying either that the thermal conductivity \(k_\textrm{th}\) is low or that the volumetric heat capacity \(\rho C_\textrm{p}\) is high. Insulating materials and polymers typically fall under this category. Because heat energy does not readily diffuse into the surroundings, these materials exhibit characteristics of localized heating. In contrast, a large value of \(\mu \) is associated with materials that have high thermal conductivity \(k_\textrm{th}\) or low volumetric heat capacity \(\rho C_\textrm{p}\). A representative example is graphene, a two-dimensional material composed of a single layer of carbon atoms arranged in a hexagonal lattice. Due to its low density and very high thermal conductivity compared to metals, it exhibits high thermal diffusivity and the characteristics of scattered heating.
In Sect. 2, we review an integral representation formula of Eq. (1) and prove Theorem 1.1. In Sect. 3, we provide a proof of Theorem 1.2 and analyze the effects of the physical parameters on the solution of Eq. (1). Section 4 presents numerical experiments that support the theoretical results of the previous sections.
[18] Let \(U \in C^2 ([0,\infty ); \Omega _h)\) be a solution to (12) with \(F \in L^{\infty } ([0,\infty ); \Omega _h)\). We assume that \(h \in (0, 1/3a)\) and let \(\varvec{\alpha }_1(h,a)\) be the smallest positive solution of (8). Then, for each \((x,x_3) \in P \times [0,h]\) and \(t\ge 0\), the solution U satisfies
where we denoted by \(\int _{P \times \{0,h\}} f(y,y_3)\textrm{d}S_y\) the sum \(\int _{P} f(y,0) \textrm{d}y + \int _{P} f(y,h) \textrm{d}y\) for an integrable function \(f: P \times \{0,h\} \rightarrow \mathbb {R}\) and \(\Vert F\Vert _{L^{\infty }([0,t])}:= \sup _{(y,y_3, s) \in P \times \{0,h\} \times [0,t]} |F(y,y_3, s)|\).
For the first solution \(\varvec{\alpha }_1(h,a)\) to equation (8), we have the following lemma.
From the above estimates, we find that \(\lim _{h \rightarrow 0} \frac{\varvec{\alpha }_1(h,a)}{\sqrt{2a/h}} =1.\) Based on this property and the integral representation of Theorem 2.1, we will investigate the effect of the thickness \(h>0\) and the material property of the film on the focused heat flux. We now discuss how the parameters affect the quality of local heating.
For a solution V(x, t) of (1), we consider the rescaled function \(U(x,t) := V(x,\frac{t}{\mu })\) which satisfies
for all \(t>s\) and \((x,y) \in P \times P\) with \(M = vol(P)\). Here, \( a \wedge b\) for \(a,b\in \mathbb {R}\) denotes the value of \(\min \{a,b\}\). Replacing t and s by \(\mu t\) and \(\mu s\) respectively, we have
If \(\mu t <M\), then \(\mu (t-s) \wedge M =\mu (t-s)\) in the integration on the right-hand side of (24). Then changing the variable s to \(t-s\) gives the desired estimate (6).
If \(\mu t >M\), we use the fact that \(\mu (t-s) \wedge M = \mu (t-s)\) for \(s \in [t - M/\mu , t]\) and \(\mu (t-s) \wedge M = M\) for \(s \in [0, t-M/\mu ]\) to decompose the integration on the right-hand side of (24) as follows:
It gives the desired estimate (7). The proof is done. \(\square \)
3 Effect of parameters on the boundary heating
In this section, we investigate how the parameters \(a>0\), \(h>0\), and \(\mu >0\) influence the solution to problem (1) in regions away from the heat flux. To this end, we utilize Theorem 1.1 to derive the estimate stated in Theorem 1.2.
Proof of Theorem 1.2
From the estimate (24), we see that for any \(t>0\), the following estimate holds:
Next we estimate the second integration on the right-hand side of (28). For this, we apply the Cauchy-Schwartz inequality to find the following inequality:
Combining this with (20) gives the desired estimate (11). The proof is finished. \(\square \)
Now we discuss the effect of the parameters a, h and \(\mu \) on the right-hand side of (37). Precisely, we derive the following principle:
The temperature in regions outside the heat flux tends to be smaller when the film thickness h and thermal diffusivity \(\mu \) are chosen smaller, and when the heat convection coefficient divided by thermal conductivity a is chosen larger.
We verify the above claim using the estimates obtained in Theorems 1.1 and 1.2 as follows.
(Impact of \(\mu \)) Fix a and h. We estimate the integral in the estimate (6). Note that
We observe that \(\frac{1}{s}\text {e}^{-\frac{\kappa R^2}{s}}\) is integrable on [0, t] and decreases as \(\mu \rightarrow 0\). Thus, we may apply the Lebesque dominated convergence theorem to yield that
It reveals that the localized heating effect becomes more transparent when thermal diffusivity \(\mu \) is chosen smaller. A numerical study and further discussion on this matter are presented in Sect. 4.1.
(Impact ofa) Fix h and \(\mu \). We know that \(\lim _{a \rightarrow \infty } \varvec{\alpha }_1(h,a) =\infty \) by Lemma 2.2. Using this we estimate the limit \(a \rightarrow \infty \) for the bound of Theorem 1.2 as follows:
This refers to the phenomenon in which thermal energy dissipates through convective heat exchange with the surroundings. If a film is in a vacuum or still air, the convective coefficient a is small, corresponding to natural convection. In contrast, if a film is exposed to a high-velocity fluid flow, the convective coefficient a will be significantly larger, leading to forced convection. As a result, the system rapidly converges to a steady state due to effective cooling. Numerical results and additional discussion on this matter are presented in Sect. 4.2.
(Impact ofh) Fix a and \(\mu \). We know that \(\lim _{h \rightarrow 0} \varvec{\alpha }_1(h,a) =\infty \) by Lemma 2.2. Furthermore, the rate of increase becomes faster as a increases. Using this we evaluate the limit \(h\rightarrow 0\) for the bound of Theorem 1.2 as follows:
This indicates that local heating becomes more effective as the film thickness h decreases. In addition, the influence of parameter h can be found in the physical significance of the volumetric heat capacity \(\rho C_\textrm{p}\). It represents the amount of thermal energy required to raise the temperature per unit volume of the material. As the thickness of the film h decreases, the material volume diminishes accordingly. Thus, less thermal energy is required to raise the temperature, resulting in rapid heating. A numerical study and further discussion on this matter is investigated in Sect. 4.3.
4 Numerical experiments
In this section, we present numerical experiments that support the arguments obtained in Sect. 3 concerning the effects of \(\mu \), a, and h on the solution to (1) in regions away from the heat source.
In order to compute the solution of (1), we will use the approximate formula (20):
The domain for our simulation is \(P=[0,100]\times [0,100]\) (mm\(^2\)), i.e., \(b=d=100\) mm. Figure 1 presents a schematic drawing of the numerical experiments, illustrating the influence of the heating parameters. Two heat fluxes are uniformly applied in a diagonal direction on a specific area of the film. The centers of the heat flux regions are at (30 mm, 70 mm) and (70 mm, 30 mm). The applied inlet heat power is 500W, and each heat flux area measures 400 mm\(^2\), forming a square with each side measuring 20 mm. To simulate the scenario depicted in Fig. 1, we compute the integral formula (48) using MATLAB. We used a basic quadrature method to calculate the integral. For the numerical evaluation of the convolution integral over the given time domain t=10s, the time was discretized with an interval of 0.1 s. Similarly, for the given spatial domain b = d = 100mm, the Green’s function kernel was computed by discretizing the space with an interval of 0.1mm.
Table 1
The values of \(\alpha _1 (h,a)\) for different values of h (mm) and a (mm\(^{-1}\))
(h, a)
\((10^{-3}, 10^{-3})\)
\((10^{-2}, 10^{-3})\)
\((10^{-1}, 10^{-3})\)
\((1, 10^{-3})\)
\(\alpha _1 (h,a)\)
1.4142
0.4472
0.1414
0.04472
(h, a)
\((1, 10^{-4})\)
\((1, 10^{-3})\)
\((1, 10^{-2})\)
\((1, 10^{-1})\)
\(\alpha _1 (h,a)\)
0.01414
0.04472
0.1413
0.4435
The values of \(\varvec{\alpha }_1(h,a)\) for various combinations of (h, a) are listed in Table 1. The value of \(\varvec{\alpha }_1(h,a)\) is determined by the film thickness h and the convection coefficient a, and changes by the amount of \(\sqrt{ha}\). This \(\varvec{\alpha }_1(h,a)\) has a significant impact on the overall tendency of the solution, as will be demonstrated in the subsequent results.
4.1 Experiment with different values of \(\mu >0\)
Figure 2 shows the temperature profiles along the diagonal line \(A{-}A'\) in Fig. 1 over time for different thermal diffusivity \(\mu \); \(\mu =1\), \(\mu =10\), \(\mu =100\), and \(\mu =1000\) [mm\(^2\)/s]. The heating simulation was run for up to 10 s, with the initial condition \(G(x,x_3 )=0\). The a was set to \(10^{-3}\) mm\(^{-1}\)], and h was set to 1mm. In the case of a small \(\mu \) value, the heat is not well diffused to surroundings, resulting in a tendency for localized heating. Conversely, with large values of \(\mu \), the heat is well distributed to the surrounding areas, at the same time, the maximum temperature in the heating zone decreases. This indicates that as \(\mu \) increases, the heating quickly converges to a static situation. Thus, larger thermal diffusivity results in faster thermal energy dispersion, a phenomenon confirmed in previous studies [22, 25].
Fig. 2
Experiment with different values of \(\mu > 0\)
Figure 3 shows the temperature profiles along the diagonal line \(A{-}A'\) in Fig. 1 at the final heating time of \(t=10\) s for varying values of \(\mu \). For very small values of \(\mu \), heat energy is localized near the heat flux inlet domains. As \(\mu \) increases, thermal energy disperses into each area, resulting in a more uniform temperature distribution.
Fig. 3
Diagonal temperature profiles with different values of \(\mu > 0\) at \(t=10\) s
4.2 Experiment with different values of \(a >0\)
Figure 4 illustrates the temperature profiles along the diagonal line \(A{-}A'\) over time for different convective coefficients divided by thermal conductivity a; \(a=10^{-4}\), \(a=10^{-3}\), \(a=10^{-2}\), and \(a=10^{-1}\) [mm\(^{-1}\)]. The \(\mu \) was set to 10 mm\(^2\)/s, and h was set to 1 mm. For small values of a, it can be observed that heat is not easily escaping to the exterior, resulting in an overall higher temperature profile compared to cases with large a values. For large values of a, thermal energy dissipates more efficiently, reducing the overall heating effect. Also, the temperature profile of the domain exhibits minimal variation over time. This indicates that the system has reached a steady-state condition, where the incoming thermal energy and convective heat transfer are balanced. This suggests that the system quickly stabilizes to a steady state when the cooling effect is strong. Figure 5 shows the temperature profiles at the final heating time of \(t=10\) s for different a values. For small values of a, the overall temperature profile is high due to a less cooling effect, but as a value increases, the overall temperature profile decreases.
Fig. 4
Experiment with different values of \(a >0\)
Fig. 5
Diagonal temperature profiles with different values of \(a>0\) at \(t=10s.\)
Fig. 6
Experiment with different values of \(h >0\)
4.3 Experiment with different values of \(h >0\)
Figure 6 presents the temperature profiles along the diagonal direction over time for different film thicknesses h; \(h=0.001\), \(h=0.01\), \(h=0.1\), and \(h=1\) [mm]. The \(\mu \) was set at 10 mm\(^2\)/s, and a at \(10^{-3}\) mm. We see from Figures 6 and 7 that the temperature outside the heating zone tends to be very small as the value of h gets close to zero. Additionally, the temperature profile within the heating zone is significantly higher for smaller values of h. This is due to the decreased volume of the film resulting from the reduced thickness, as mentioned above. The volumetric heat capacity \(\rho C_\textrm{p}\) signifies the heat energy required to increase the temperature per unit volume. As the thickness of the film decreases, the heat energy required to raise the temperature also decreases. Similarly, as the film thickness increases, more thermal energy is required to raise the temperature. Additionally, a significant amount of energy will dissipate to other regions. Consequently, compared to the case with thinner films, a substantially lower temperature profile can be observed over the same time period. Figure 7 shows the temperature profiles for different h values at the final heating time of \(t=10\) s. When the h value is small, the heating occurs more efficiently, hence an overall higher temperature profile than larger h value cases. However, as the h value increases, more thermal energy is required to raise the temperature, leading to a decrease in the overall temperature profile.
Fig. 7
Diagonal temperature profiles with different values of \(h>0\) at \(t=10\) s
5 Conclusion
The use of thin films in engineering applications has grown significantly in recent years, particularly in microelectronics packaging, where film thicknesses continue to decrease. This reduction in thickness poses new challenges for thermal management, especially in accurately measuring and controlling temperature. Motivated by these challenges, this work presents a mathematical analysis of the influence of physical parameters on local surface heat flux of thin films. We prove that the temperature outside the heating zone tends to remain low when the film thickness h and thermal diffusivity \(\mu = k_\textrm{th}/(\rho C_\textrm{p})\) are small, and the heat convection coefficient a is large. Based on an approximate integral formula using the Green’s function obtained in Lee and Choi [18], this paper provided new proofs and physical discussion. This mathematical discussion has been validated through numerical simulations, which analyze the temperature distribution under localized surface heat flux. It was confirmed that as thermal diffusivity decreases, temperature tends to concentrate in localized areas, which is a well-known phenomenon. In addition, it was observed that as the thickness drastically decreases, the concentration of heat in the local region subjected to heat flux becomes more pronounced. This highlights the importance of efforts to increase diffusivity in engineering applications where the thickness of thin films is reduced. Furthermore, it was found that the convection coefficient is also shown to play a critical role in modulating localized temperature rise. The study provides physical intuition regarding the influence of physical variables under localized heating conditions of thin films. While the current analysis focuses on a single-layer film, future research should extend to multi-layer thin films and incorporate experimental validation to support practical applications as a key focus.
Acknowledgements
Woocheol Choi and Eun-Ho Lee are co-corresponding authors. This work was supported by the National Research Foundation of Korea (NRF) through Grants (Nos. 2021R1C1C1007946 and RS-2024-00336077) funded by the Korean Government, and by the Technology Innovation Program (or Industrial Strategic Technology Development Program) (RS-2023-00236091, Hybrid bonding technologies for 3D package interconnects) funded by the Ministry of Trade, Industry & Energy (MOTIE, Korea) (1415187584).
Declarations
Conflict of interest
The authors state that there is no conflict of interest.
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