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Predicting Physical Properties of Nanofluids by Computational Modeling Natalia Sizochenko, Michael Syzochenko, Agnieszka Gajewicz, Jerzy Leszczynski, and Tomasz Puzyn J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b08850 • Publication Date (Web): 27 Dec 2016 Downloaded from http://pubs.acs.org on December 28, 2016

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The Journal of Physical Chemistry

Predicting Physical Properties of Nanofluids by Computational Modeling Natalia Sizochenko1,2, Michael Syzochenko2,3, Agnieszka Gajewicz1, Jerzy Leszczynski2, Tomasz Puzyn1* 1

Laboratory of Environmental Chemometrics, Faculty of Chemistry, University of Gdansk, Wita

Stwosza 63, 80-308, Gdansk, Poland 2

Interdisciplinary Center for Nanotoxicity, Department of Chemistry, Jackson State University,

1400 J. R. Lynch Street, P. O. Box 17910, 39217, Jackson, MS, USA 3

Department of Computer Science, Jackson State University, 1400 J. R. Lynch Street, P. O. Box

17910, 39217, Jackson, MS, USA

1 ACS Paragon Plus Environment


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ABSTRACT: The focal point of the current contribution was to develop global quantitative structure-property relationship (QSPR) models for nanofluids. Two target properties: thermal conductivity and viscosity of nanofluids were thoroughly investigated. Under this investigation a new database of thermal conductivity and viscosity of nanofluids (more than 150 data points) was created. A hierarchical system of molecular representation reflecting features of nanoparticle’s structure at the different levels of organization was introduced. Also sizedependent, volume-dependent and intensive parameters were calculated. The model for thermal conductivity is characterized with determination coefficient R2 = 0.81 and root mean squared error RMSE = 0.055; model for viscosity: R2 = 0.79 and RMSE = 0.234. Developed models are in agreement with modern theories of nanofluids behavior. Size- and concentration-related behavior of target properties were discussed. Findings suggest that the increase of surface area ratio, interfacial layer thickness and decrease of nanoparticles size lead to thermal conductivity and viscosity increase. Thermal conductivity and viscosity increase with an increase of weighted fraction-dependent parameters. Up to date, reliable theoretical models were created only for a single type of nanoparticles. In the current article, developed models can simultaneously predict the thermal conductivity and viscosity in an effective way using both size and volume concentration of nanofluid.


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Introduction Nanofluids are engineered colloids of nanoparticles (NPs) dispersed homogenously in a base fluid.1 Transport, thermophysical and other related properties of nanofluids differ from properties of base fluids.2-4 For instance, addition of small fractions of NPs can cause significant enhancement of thermal conductivity. Many experimental and theoretical works reported equations for thermal conductivity and viscosity changes.2-3, 5-15,

16-18

Existing equations explain

the influence of nanofluid’s viscosity and thermal conductivity using different parameters, e.g. type of base fluids, NP volume fraction, particle size, particle shape, temperature, surface charge, pH value, particle nature, Brownian motion of NPs, effect of clustering, monolayer and dispersion techniques.1,

4, 19-20

Almost all available studies report size-dependent and/or

concentration-dependent thermal conductivity and viscosity focusing on one type of nanofluids. However, there is a lot of controversy over experimental findings and theoretical models. So far, no satisfactory mathematical models have been proposed to describe the thermal response of a nanofluid. One of the most promising and popular techniques to develop mathematical dependencies between target properties and features of the chemical structure is the Quantitative StructureProperty/Activity Relationships (QSPR or QSAR) approach.21 In the case of nano-sized compounds, this technique is often referred to as “nano-QSPR” or ”nano-QSAR”.22-23 A large number of nano-QSAR/QSPR models has been developed for inorganic nanoparticles since 2009.23 Despite the urgency of this task and promising results of the existing nano-QSAR/QSPR models, there is no universal methodology that provides a unified description for a variety of nanoscale structures. Moreover, direct description of dispersed-phase particles is possible only



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using molecular dynamics models,8 but these methods are not suitable for fast predictions for screening purposes. There are several studies on the influence of the chemical structure on thermal conductivity and viscosity of nanofluids by using the QSPR approach.24-25 All of them employed simplified representation of the nanofluid structure. For example, in our previous contribution, nanofluids structures were represented by so-called “liquid drop” model.25 However, applicability of the model lacked (1) diversity (only 5 types of NPs were considered) and (2) volume fraction (NPs concentration lays within 0.01 - 0.55 %). Also only thermal conductivity was considered as a target property.25 The current study is aimed at a detailed analysis of the thermal conductivity and viscosity using hierarchical combination of newly developed descriptors, which reflect nanofluid structure at the different levels of organization. This contribution aims at the creation of a unified expert system, which could predict thermal conductivity and viscosity values for different types of nanofluids at different concentrations and sizes. In this paper, the values of thermal conductivities and viscosities for nanofluids, predicted with the proposed computational model are compared with the experimental data reported in the literature.

Experimental Data As it was discussed in previous contributions,25-26 physical properties of nanofluids may depend on the type of the base fluid as well as on the size of NPs. The original experimental data on thermal conductivity and viscosity were gathered from literature (see Supporting Information, Table S1) in the framework of NanoBRIDGES project (FP7-PEOPLE-2011-IRSES, grant agreement #295128). The gathered dataset was presented in current form as a part of


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NanoBRIDGES report. In the current investigation applicability domain of the model is restricted to spherical and near-spherical NPs at temperature range of 21-25 0C (the most popular temperature range among collected research articles). We have collected the largest available dataset, consisting of 100 nanofluids for which the experimentally measured thermal conductivity data were available (k) and 69 nanofluids for which the experimentally measured viscosity data were available (η). The new datasets are larger than the dataset, collected in our previous contribution (thermal conductivity, 23 data points).25 In our previous contribution the database consisted of nanofluids which contained metals (Cu, Fe) and metal oxides (Al2O3, TiO2, ZrO2) NPs. In current work, we report a highly diverse database which consists of experimental data for different types of nanofluids: metals (Au, Ag, Cu, Fe), metal oxides (Al2O3, CuO, Fe2O3, TiO2, ZrO2) and silica-derived (SiO2, SiC) nanofluids dispersed in water. The thermal conductivity ratio (k) was defined as the ratio of nanofluid’s thermal conductivity to thermal conductivity of pure fluid (water). The viscosity ratio (η) was defined as the ratio of nanofluid viscosity to viscosity of pure fluid (water). Additionally, the experimental data on the average NP size (SA) and the volume fraction of NPs ( ) for each nanofluid was extracted. Structures of NPs, target properties (the endpoints) and related experimental values are presented in SI (Tables S2 and S3).

Theoretical Methods Descriptors Several levels of the structure representation were applied. First, chemical diversity of the collected database was reflected by simple descriptors based on the periodic table (PTD): heat capacity (heat), mass density (ρ), molecular mass (M), indicator (1 for yes/0 for no) for metal



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oxides of 4th or 5th period metal oxides (PTD-1), number of cations in metal oxides composition (PTD-2 for 4th period, PTD-3 for 5th period, PTD-5 for 3th period).31-32 Next, size-dependent descriptors derived from the “liquid drop” model (LDM) were utilized to describe the features of pristine nanoparticles.26 The scheme of LDM calculations was discussed in our previous contributions.25-26 It was demonstrated, that LDM encodes possible interactions between NPs using simplified geometrical representations of the particles at the nano-level. In LDM, the basic elements are densely packed and the density of the “liquid drop” is equal to the mass density (so-called Wigner-Seitz radius, eq. 1):

=

(1)

where: M - molecular mass, ρ - mass density. LDM approximation assumes, that probable NP’s shape is a sphere. Based on this assumption, the number of elementary particles in a nanoaggregate is defined as follows (eq. 2):

n =

(2)

where: r0 – the mean radius of a NP. Using equation 2, nanoparticle’s surface area (F, eq. 3) and the surface-area-to-volume ratio (SV, eq. 4) could be calculated: F = 4 SV =

(3)

! #$%! %!

= #$%!

%! & '$%#!

(4)

Unique properties of nanofluids are affected by the layering of liquid molecules at the solid-particle surface (so-called, interfacial layer or nanolayer).27-28 Interfacial layer plays the role of a bridge between a solid NP and the base liquid. NPs have highly ordered structures, whereas base liquids are less organized. The interfacial layer is more ordered than the bulk 6 ACS Paragon Plus Environment

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liquid, but less ordered than the solid structure of NP. Thus, to reflect, specific adsorbed layer between surface of NPs and fluid, the interfacial nanolayer model (INM) was used (Figure 1).2729

Several models were developed to describe features of interfacial nanolayer, for example

thickness. Thickness of this layer (h, nm) is highly dependent on the temperature and can be described by the following equation (eq. 5):29 ( = 0.01 ∙ - . 273 ∙ 23.4

(5)

where 2 – size of single NP [nm].

Figure 1. Structure of nanoparticle in the base fluid. Molecules of the liquid (2) can form specific, highly ordered layer (3) near the nanoparticle surface (1).

Let us refer to the LDM- and INM-based descriptors as unweighted descriptors. Unfortunately, both types of descriptors are not able to express the effects related to changes of the volume fraction of NPs.3 When the value of the target property changes with the concentration of its components, a weighting procedure could be applied to describe concentration-dependent changes.30 Weighted descriptors are calculated as concentrationweighted sums using the descriptor value and volume fraction of NP (eq. 6): A6 = ∙ 7,

(6)



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where Aw - is the weighted descriptor, – volume fraction of NPs, A – unweighted descriptor. It should be noted that in this case, each NP is defined by several points (several property values) corresponding to the different concentration values. Experimentally measured structure characteristics, such as size and concentration of NPs were also considered as descriptors. All calculated descriptors are provided in the electronic SI (Tables S4, S5, S6).

Model development and validation procedures For the purpose of validation, the initial dataset was split between training and test sets.21 Each training set covers ~ 80% of the initial dataset; related test set covers the remaining ~ 20% (see SI, Tables S7, S8). Nanofluids from the training set were structurally diverse enough to cover the whole descriptors space of the overall data set. Values of thermal conductivity were ranged in order to split datasets evenly. Each type of nanofluid (metal, metal oxide, etc.) was included in both training and test sets.21 Chemical diversities of nanofluids in the training and test sets were similar. After calculating an initial pool of descriptors, the descriptors with zero-variance and highly internally correlated (|r| > 0.80) ones were eliminated and the procedure of z-scaling (standardization) was performed. The models were developed using the Weka software package33 and its integration workflow plan for KNIME 2.11. For the purposes of QSPR modeling the M5P classifier was applied.33-34 M5P methodology combines a conventional decision tree technique with the possibility of including linear regression functions at the nodes. In the training set, each NP is specified by the fixed set of descriptors and has an associated target value (here: thermal


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conductivity or viscosity). Tree-based M5P models are constructed by the divide and conquer method. At the first step tree model is generated and specific tree nodes are specified. At the next step, multivariate linear model is constructed for the training samples at each node obtained at first step. The statistical fit of the QSPR models and their predictive ability were assessed using different internal and external validation strategies. The goodness-of-fit of the models was estimated by the quality metric R2, as well as external validation by metric R2test.35 Additional rm2 2

metrics for the training and test set data ( rm and ∆rm2) were estimated to determine the closeness between the values of predicted and corresponding observed.35 Moreover, for each model we calculated the root-mean-square error (RMSE) both for test and training sets. One of the OECD principles for model validation requires defining the Applicability Domain (AD) for the QSPR models.21 Reliable predictions are impossible only for chemicals structurally similar to the training compounds used to build the QSPR model. As in present work we used several types of chemically equal compounds, we suggest, that current models are applicable only for similar compounds in the same experimental conditions.

Results and Discussion Thermal conductivity model 2

The model is characterized by R2 = 0.81, RMSE = 0.055, rm = 0.76, ∆rm2 = 0.15 for the 2

training set; R2 = 0.70, rm = 0.67, ∆rm2 = 0.20 and RMSE = 0.077 for the test set. Dataset splitting, experimental and predicted values for the thermal conductivity model are summarized in SI (Table S7). Satisfactory values of the correlation coefficients and low values of the prediction errors confirm both accuracy of fitting, and predictive ability of the developed model. 9 ACS Paragon Plus Environment


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Graphical representation of the correlation plot is presented in Figure 2.

Figure 2. Observed (experimental) vs. predicted values of thermal conductivity (k). Dots represent training set, squares – test set. It can be seen that some values are predicted with lower accuracy than others. Thermal conductivity ratios for nanofluids which contained larger NPs than usual, or high concentrations (higher than 1 %) were predicted with > 10 % error. For instance, ZrO2 (20 nm, 1 - 2 %, Table S2, nanofluids № 13 and № 14), TiO2 (50 nm, 1 - 2 %, Table S2, № 13 and № 14), Al2O3 (33 nm, 4 – 5 %, Table S2, № 49 and № 50), Cu (75 nm, Table S2, № 31) CuO (5 nm, 5 %, Table S2, № 55; and 23 nm, 10 %, Table S2, № 79). Nanofluid which contained CuO of 23.9 nm, 2.5 % (Table S2, № 63) was highly overestimated and thus excluded from final model. Mathematically, M5P model for thermal conductivity of nanofluids could be presented as 10 ACS Paragon Plus Environment

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a series of if-then-else rules (Figure 3).

Figure 3. M5P tree for thermal conductivity (k in %). As one can see (Figure 3), the developed model contains nine descriptors, from which the collection of rules included four pure descriptors: density (ρ), molecular weight (M), number of 5th period cations in chemical formula of metal oxide NP (PTD-3), presence/absence of metal



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oxides of periods 4 and 5 (PTD-1); two weighted descriptors: number of NPs per volume fraction (nw) and heat capacity per volume fraction (heatw). Node equations included two unweighted descriptors – volume fraction of NPs ( ) and number of 4th period cations in chemical formula of metal oxide NP (PTD-2) and two weighted descriptors number of NPs per volume fraction (nw) and INM per volume fraction (hw). Relative importance values (RI), namely – the percentage of involvement of each descriptor – are presented in Table 1. Table 1. List of descriptors and their relative importance (RI, in %) in the thermal conductivity model Descriptor

Symbol

Descriptor type RI, %

volume fraction of NPs

experimental

3.5

molecular weight

M

PTD

14.0

presence/absence of metal oxides of periods 4 and 5

PTD-1

PTD

15.0

number of 4th period cations in chemical formula of metal oxide NP

PTD-2

PTD

4.0

number of 5th period cations in chemical formula of metal oxide NP

PTD-3

PTD

4.0

density

ρ

PTD

30.0

number of NPs per volume fraction

nw

weighted LDM

8.5

heatw

weighted PTD

5.0

hw

weighted INM

15

heat capacity per volume fraction interfacial layer thickness per volume fraction

Technical model node for KNIME is presented in SI. Installation process is presented on official website (https://www.knime.org/). Potential users can install this node and use it to make own prediction. Alternatively, if-then-else rules could be applied directly. For instance, for NPs, which mass density (ρ) is less than 4580, the first equation from if-then-else model should be applied (eq. 7): 12 ACS Paragon Plus Environment

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k = 7.6103 · PTD-1 + 2.8122 · hw + 0.9347 (7) In case if mass density (ρ) is higher than 4580, and molecular mass is lower than 59.75, the second equation from if-then-else model should be applied (eq. 8): k = -1.4912 · + 15.0476

(8)

This procedure should be repeated until suitable rules are found.

Viscosity model

The developed model for viscosity is characterized with quite good statistical parameters: 2

2

R2 = 0.79, rm = 0.73, ∆rm2 = 0.088 and RMSE = 0.234 for the training set; R2 = 0.78, rm = 0.72, ∆rm2 = 0.14 and RMSE = 0.244 for the test set. Dataset splitting, experimental and predicted values for the viscosity model are summarized in SI (Table S8). The plot of experimental versus predicted values is shown on Figure 4.



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Figure 4. Observed (experimental) vs. predicted values of viscosity (η). Dots represent test set, squares – training set. Viscosity of nanofluids which contained Al2O3 at high concentrations (higher than 5 %) were usually overestimated or underestimated with > 23 % error. For instance, nanofluids (Table S3) № 5 (5 %), № 6 (6 %), № 12 (9 %), № 15 (8.5 %) and № 16 (12.2 %). Next, in the case of SiC (Table S3, № 66-69) predictions were similar. This issue could be related to non-metallic behavior of selected nanofluids (All other NPs were metals or metal oxides). Similar situation was observed for nanofluids with CuO at 33 nm (Table S3, № 53-59). At the same time, predictions for other nanofluids with CuO (Table S3, № 60-65) were characterized with high variation (Pearson correlation coefficient 0.93).


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Mathematically, M5P model for viscosity of nanofluids could be presented as a series of if-then-else rules (Figure 5).

Figure 5. M5P tree for viscosity. The developed model for viscosity contains seven parameters. Collection of if-then-else rules included two pure descriptors: number of 3rd period cations in chemical formula of NP (PTD-4) and interfacial nanolayer thickness (h); three weighted descriptors (number of NPs per volume fraction, nmix; heat capacity per volume fraction, heatw; surface ratio, Fw). Node equations included pure descriptors of presence/absence of TiO2, (PTD-5) and three weighted descriptors: density per volume fraction (ρw), number of NPs per volume fraction (nw) and heat 15 ACS Paragon Plus Environment


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capacity per volume fraction (heatw). Relative variable importance of descriptors is presented in Table 2. Technical information about KNIME node are presented in SI. Table 2. List of descriptors and their relative importance (RI in %) in the viscosity model Descriptor

Symbol Descriptor type RI, %

number of 3rd period cations in chemical formula of NP

PTD-4

PTD

29.0

presence/absence of TiO2

PTD-5

PTD

8.0

heat capacity per volume fraction

heatw

weighted PTD

8.0

density per volume fraction

ρ

weighted PTD

8.0

surface area ratio per volume fraction

Fw

weighted LDM

19.0

number of NPs per volume fraction

nw

weighted LDM

29.0

interfacial layer thickness

hw

INM

15.0

Interpretation of developed models

The models developed within this study represent physical properties of nanofluids as functions of experimentally measured and calculated descriptors. As it is shown in Tables 1-2, there are several ways of combining important descriptors into groups. Let us refer to these groups with fundamental physical concepts. One important physical parameter, which plays a major role in nanofluid thermal properties and stability, is the size of NP.2-5 In most cases, thermal conductivity and viscosity could linearly or non-linearly increase as particle size decreases.4 In the developed models this outcome is supported by the size-dependent LDM- and INM-based descriptors, namely surface area ratio, interfacial layer thickness and number of NPs. As is was previously mentioned “liquid drop” model-derived and interfacial nanolayer model derived descriptors are size-dependent. As developed M5P model is non-linear, both effects of potential linear and non-linear decrease and


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increase are reflected. Cumulative impact of these descriptors is 23.5% for the thermal conductivity model and 63% for the viscosity model. The other important empirical finding is NPs fraction-property response of nanofluids.1-10 Thermal conductivity and viscosity increase with volume concentration. As weighted descriptors are fraction-dependent, these fundamental relationships are clearly reflected in the developed models. Cumulative relative influence for the fraction-related descriptors is 32% for the thermal conductivity model and 64% for the viscosity model. On the other hand, with the decrease in volume fraction, effective volume of the cluster increases, hence the enhancement of thermal conductivity and viscosity. Clustering of NPs can create paths of lower thermal resistance and enhance thermal conductivity significantly. If one has an aggregated cluster of closely packed spherical particles, ~ 25% of the volume is filled with liquid in the space between particles, which increases the effective volume of the highly conductive region by ~ 30%. The last group of properties – intensive properties of NPs (here: PTD) contribute 72% for the thermal conductivity model and 53% for the viscosity model. An intensive property is a bulk property, meaning that it is a physical property of a system that does not depend on the system size or the amount of material in the system (here: periodic-table based). Comparing our results with existing models (here: models for nanofluids with water as a base fluid) we have found that every research group compared their data with existing theoretical models only for single type of NPs (using different volume concentration).1,3,4 For instance, Duangthongsuk et al.,6 Lee et al.,13 Li et al.,15 and Paul et al.19 measured the target properties and

built relationships of them with volume fraction or compared their own results with results of other authors. Kang et al.,11, Lee et al.,14 and Xie et al.,17 measured target properties and compared their results with the Hamilton-Crosser model.9 Kang et al.,11 found that the Hamilton-



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Crosser model predicts properties of Ag nanofluids with lower values in comparison to the experimental data; this model works well for SiO2 nanofluids. However, Kang et al.,11 does not compare Ag and SiO2 nanofluids simultaneously. Lee et al.,14 demonstrated that his experimental data is in agreement with the Hamilton-Crosser model for Al2O3 nanofluids, but gave inadequate results for CuO nanofluids. Beck et al. 5 used a simple equation (k = 4.4134 ) to measure thermal conductivity for

Al2O3 nanofluids. R2 for this equation was between 0.90 and 0.93. Unfortunately, this equation was not tested for other types of nanofluids and is not applicable to NPs of different sizes. Therefore, our models are the only models, which can simultaneously predict the thermal conductivity and viscosity in an effective way using both size and volume concentration of nanofluid.

Conclusions In the present study we have successfully applied the computational modeling to build theoretical models of thermal conductivity and viscosity of nanofluids. We introduced a new hierarchical system of structure representation which allows reflecting features of nanofluids’ at the different levels of organization: from intensive properties (as period of metal, density, etc.) to nanoparticles size, concentration and thickness of interfacial layer. Existing models of thermal conductivity and viscosity of nanofluids can predict only sizeor concentration-dependent behavior. On the contrary, models developed in the framework of hierarchical approach, make an allowance for predictions of thermal conductivity and viscosity for nanofluids that contain NPs of different sizes, concentrations and types (metal, silica- or metal oxide). KNIME nodes for developed models are available for public use.


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We have developed statistically significant and externally predictive quantitative structure-property relationships models. Models were obtained using the largest available dataset of structurally diverse nanofluids, including a variety of concentration and size classes. From this study it is clear that the best models included a combination of different types of descriptors. Developed models are in agreement with the fact, that viscosity and thermal conductivity depend on many parameters such as particle volume fraction, particle size, particle size distribution and thickness of interfacial layer. From the performed calculations we found the descriptors that are highly correlated to the observed viscosity and thermal conductivity. These descriptors are mainly related to the volume fraction and size of the nanomaterials. It was found that size-dependent “liquid drop” model and the interfacial nanolayer model derived descriptors (surface area ratio, interfacial layer thickness, number of NPs) have positive influence on thermal conductivity. Weighted fraction-dependent descriptors reflected an increase in thermal conductivity and viscosity with increase of volume concentration. Applicability of the models is limited to spherical nanoparticles. We believe, that the obtained models reflect substantial factors concerning the physical properties of nanofluids. The developed models could help in the future with the development of suitable candidate materials.

AUTHOR INFORMATION Corresponding Author *Phone: + 48 (058) 523-5248 Dr. Tomasz Puzyn, e-mail: [email protected] Author Contributions N.S. and M.S. contributed equally. N.S. and M.S. collected data, carried out calculations, processed the data, developed code and analyzed results. N.S., M.S. and A.G. drafted and revised 19 ACS Paragon Plus Environment


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the manuscript. J.L. and T.P. supervised the research, coordinated the study and revised the manuscript. All authors have read and have given approval to the final version of the manuscript. Notes The authors declare no competing financial interests. ACKNOWLEDGMENTS The authors thank the support from the NSF CREST Interdisciplinary Nanotoxicity Center: grants EPSCoR 362492-190200-01\NSFEPS-0903787 and N00014-13-1-0501. This research has received funding from the European Union Seventh Framework Program, NanoPUZZLES project (FP7/2007-2013, grant agreement #309837) and the European Commission through the Marie Curie IRSES program, NanoBRIDGES project (FP7-PEOPLE-2011-IRSES, grant agreement #295128). SUPPORTING INFORMATION Collected experimental data, values of calculated descriptors and KNIME code are provided in the electronic Supporting Information.

ABBREVIATIONS NP nanoparticle; LDM “liquid drop” model; QSAR Quantitative Structure-Activity Relationships; QSPR Quantitative Structure-Property Relationships; RMSE the root-meansquare error.

REFERENCES 1.

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