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The density of the base oil + dispersant mixtures was measured using a variable-volume view cell described in previous publications [11, 12]. The view cell consists of a main body with two sapphire windows on opposite sides in order to view the internal contents of the cell, and a variable-volume part that houses a movable piston. The piston position is monitored with an LVDT (linear variable differential transformer). The position of the piston is altered using a back-pressure fluid (such as ethanol) activated by a motorized pressure generator which allows changing the internal volume and thus the pressure in the cell in a controlled manner. Measurements are taken along isothermal pressure scans that occur at a rate of approximately 0.1 MPa·s−1. The sampling rate is approximately one data point for every 0.5 s. The fluid pressure is monitored with a Dynisco flush mount pressure transducer that has an uncertainty of ± 0.07 MPa. (All uncertainties stated in this study are standard uncertainties). Temperature in the cell is monitored with an RTD 100 platinum resistance temperature detector which has an uncertainty of ± 0.1 °C. The internal volume of the cell was calibrated by measuring the density of ethanol and comparing the results with the literature values [13, 14]. The internal volume was calibrated within ± 0.82% uncertainty.
Viscosities were measured using a high pressure rotational viscometer designed in the lab that has been described in our earlier publications [12, 14]. The inner rotating shaft has a magnet embedded at the top which allows the shaft to be magnetically coupled to a torque transducer outside of the viscometer cell without affecting the high pressure sealing arrangement of the cell. Torque is measured with an uncertainty of ± 0. Nm using a Thermo Scientific Instruments, Haake viscometer 550 torque transducer. The cell is pressurized using movable pistons activated by a pressure generator using ethanol as the back-pressure fluid. Data were collected at each temperature by changing the pressure at a rate of about 0.2 MPa·s−1, with a sampling rate of one point for every 0.5 s. This pressure change was, however, achieved using a manually operated pressure generator, leading to some small variations in the rate of pressure change in different scans. The pressure of the cell is monitored with an Omega PX91N0-60KSV pressure transducer which has an uncertainty of ± 0.16 MPa. The temperature is monitored with an RTD-100 temperature detector which has an uncertainty of ± 0.1 °C. The cell has two pairs of sapphire windows allowing the visual verification of the rotation of the shaft and the state of the oil.
The viscometer was calibrated at pressures up to 50 MPa and temperatures up to 373 K using tris(2-ethylhexyl) trimellitate ester as a viscosity standard [14, 15]. With this viscometer, viscosities are determined with an uncertainty of 2.9 % in the range from 13.9 mPa*s to 270 mPa*s.
The density results were correlated with the Sanchez-Lacombe EOS which is based on a lattice-fluid model given by the following equation [16]:
$${\widetilde{\rho }}^{2}+\widetilde{P}+\widetilde{T}\left(\text{ln}\left(1-\widetilde{\rho }\right)+\left(1-\frac{1}{r}\right)\widetilde{\rho }\right)=0$$ (1)in which \(\widetilde{\rho }\), \(\widetilde{P}\), and \(\widetilde{T}\) are the reduced density, pressure, and temperature given by ρ/ρ*, P/P*, and T/T*, respectively, and r is the size parameter representing the size of lattice sites occupied. Further details on this equation were previously published [14].
From the Sanchez-Lacombe equation, isothermal compressibility, isobaric expansivity, and internal pressure can be determined using the following equations [11, 17].
$${\kappa }_{T}=-\frac{1}{V}{\left(\frac{\partial V}{\partial P}\right)}_{T}=\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial P}\right)}_{T}=\frac{\widetilde{P}{\widetilde{v}}^{2}}{P\left(\widetilde{T}\widetilde{v}\left[\frac{1}{\widetilde{v}-1}+\frac{1}{r}\right]-2\right)}$$ (2) $${\beta }_{P}=\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_{P}=-\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial T}\right)}_{P}=\frac{1+\widetilde{P}{\widetilde{v}}^{2}}{T\left(\widetilde{T}\widetilde{v}\left[\frac{1}{\widetilde{v}-1}+\frac{1}{r}\right]-2\right)}$$ (3) $$\pi ={\left(\frac{\partial U}{\partial V}\right)}_{T}=T{\left(\frac{\partial P}{\partial T}\right)}_{V}-P=T\left(\frac{{\beta }_{P}}{{\kappa }_{T}}\right)-P$$ (4)Accounting for propagated uncertainty from pressure, temperature and density measurements, the derived quantities can be determined within approximately 2% uncertainty for isothermal compressibility and isobaric expansivity values and 3% uncertainty for internal pressure values.
Viscosity and density were correlated using both a free-volume correlation and density-scaling approach. The free-volume correlation used was the Allal et al. [18] relationship:
$$\eta ={\eta }_{0}+\frac{\rho l(\alpha \rho +\frac{PM}{\rho })}{\sqrt{3RT M}}\text{exp}\left[C{\left(\frac{\alpha \rho +\frac{PM}{\rho }}{RT}\right)}^\frac{3}{2}\right]$$ (5)This equation correlates viscosity (η) with density (ρ), temperature (T), and molecular weight (M). The parameter η0 is the dilute gas term, or the viscosity contribution assuming the density is null. For a dense fluid, this term is negligible. As for the other parameters, l is the characteristic dissipative length for molecular energy, α is related to the energy barrier that a molecule must overcome to diffuse, and C is the free-volume overlap. The characteristic dissipative length parameter l is equal to L2/bf in which L is a characteristic length and bf is the dissipation length of a given quantity of energy.
The density-scaling correlations were based on the following relationship [19]:
$${\upeta } = {\text{A exp}}\left[ {\left( {{\text{B}}\rho^{\gamma } } \right)/{\text{T}}} \right]$$ (6)In this equation A, B, and γ are fitted parameters. The parameter γ is a constant for a given substance [20] and is reported to be lower in systems in which the viscosity has more dependence on temperature and higher in systems in which viscosity has greater dependence on density. On a microscopic scale, this density-scaling constant is linked to potential energy as it has been reported to correlate with the exponent n in the Lennard–Jones type potential energy relationship [21]:
$$U\left(r\right)= \varepsilon \left[{\left(\frac{\sigma }{r}\right)}^{n}-{\left(\frac{\sigma }{r}\right)}^{6}\right]$$ (7)in which σ is the diameter of the molecule which may be envisioned as a particle, r is the distance between them, and n is a constant of the material. The higher the n value is, the higher the potential energy between two interacting molecules, and the higher the gamma parameter.
The parameters A and B in the density-scaling equation have been shown to be correlated to the size and sphericity of the molecules [22]. Specifically, A has been found to increase with increasing chain length and B has been found to decrease with increasing chain length and nonsphericity of the molecules.
Figure 3 shows the density data for base oil Ultra S4 modified with 1, 2 and 5 wt % of lower molecular weight (LMW) (950 g·mol−1) uncapped (Unc) dispersant. These figures also include results of the correlations of the density with the Sanchez-Lacombe equation of state. The characteristic parameters ρ*, P*, and T* that describe these correlations for the base oil modified with dispersants or PIB additions are given in Tables 2, 3, including the oils modified with higher molecular weight (HMW) ( g·mol−1) uncapped as well as the low and high molecular weight capped (Cap) dispersants. The trends in the changes in these parameters in different oil systems are graphically illustrated in Figs. 4, 5, 6, 7, 8, 9. As shown, the addition of either the LMW or HMW versions of uncapped or capped dispersants to Ultra S4 leads to a decrease in P* and an increase in ρ* and T* from that of the original base oil. A more consistent increase in the closed-pack density, ρ*, occurs with the addition of the higher molecular weight additives.
Figure 10 is a bar plot showing the trends in the density at 298 K and 10 MPa for these oils. These trends were similar at different temperatures and pressures. Compared to the base oil, highlighted by the dashed black line in the figure, all additives are found to lead to higher densities. The largest increases are observed with the HMW dispersant at 2 and 5 wt % addition levels. Capped dispersants tend to lead to higher densities compared to their uncapped forms. It is possible that the uncapped dispersants are not able to pack as tightly as the capped dispersants, with their polar group being uncapped, creating more repulsive moieties compared to the nonpolar base oil molecules.
Additional density comparisons for dispersants as well as the PIB of a similar molecular weight at 1, 2 and 5 wt % addition levels are provided in Supplementary Figs. S1–S6.
Using Eq. 2 and the characteristic parameters given in Tables 2 and 3 isothermal compressibility of each oil sample was evaluated. Figure 11 is a comparative bar diagram showing the change in isothermal compressibility of these oil samples at 298 K and 10 MPa. These trends also remained similar at different pressures and temperatures. Almost all of the additives cause an increase in compressibility of the base oil. In general, a higher increase in compressibility is observed with the PIBs, which lacks polarity, or steric groups, and therefore are able to pack more tightly together than the dispersants under pressure. In the case of 1 and 2% LMW PIB additions, compressibilities were lower compared to the LMW dispersants at these addition levels.
Additional isothermal compressibility comparisons are provided in Supplementary Figs. S7–S12 at 298 K and 398 K as a function of pressure.
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Using Eq. 3 and the characteristic parameters given in Tables 2–3, isobaric expansivity of each oil sample has been evaluated. Figure 12 is a bar plot showing the trends in isobaric expansivity at 298 K and 10 MPa. The expansivity of the base oil tends to decrease with addition of dispersants, while the addition of PIB leads to an increase. The presence of polar and bulkier groups in the dispersants appears to prevent the molecules from expanding outwards. Additional comparisons of the isobaric expansivity of the oils modified with different dispersants and a PIB of similar molecular weights are provided in Supplementary Figs. S13–S18.
The internal pressure is evaluated using Eq. 4 and the parameters given in Tables 2 and 3.
Figure 13 is a comparative bar plot showing that the internal pressure decreases to a similar level with all additives, except for the LMW PIB where it is higher for the 1 and 2 wt % addition levels. It is possible that at these concentration levels, since the low molecular weight PIB is a non-polar hydrocarbon just like the base oil, it has little effect on the packing of the base oil until higher concentration levels are reached or higher molecular weights are involved. Additional internal pressure comparisons between dispersants and PIBs of a similar molecular weight at 1, 2 and 5 wt % are presented in Supplementary Figs. S19–S24.
Figure 14 shows the viscosity data for Ultra S4 modified with the LMW uncapped dispersants at all addition levels over a pressure range from 10 to 45 MPa at isotherms of 298 K, 323 K, 348 K, and 373 K.
In Figs. 15, 16, 17, viscosities of the modified Ultra S4 base oil are compared for LMW PIB, LMW uncapped and LMW capped dispersants at 1 wt % addition levels at all isotherms explored as well as select isotherms for the 2 wt % addition level. The viscosity results for the remaining 2 and 5 wt % addition comparisons are provided in the Supplementary Figs. S25–S27.
One of the striking differences between the oil samples is displayed by the oil modified with the 1 wt % LMW uncapped dispersant at 323 K shown in Fig. 15. With this dispersant, the increase in viscosity is greater than that with the 1 wt % addition of either LMW PIB or LMW capped dispersant. As shown in Fig. 17, at 2 wt %, the addition of LMW capped dispersant leads to the smallest increase in viscosity at 298 K and 323 K. At 5 wt % addition, the increase in viscosity becomes similar between all samples at all isotherms. It seems that at only a 1 wt % addition, the LMW uncapped dispersant is able to disrupt the molecular packing organization and create more viscous forces at lower temperatures, while a similar increase in viscosity is not created with the addition of other additives until 2 wt %. This is likely due to the polarity of the uncapped amines in the uncapped dispersant compared to the less polar PIBs or the capped dispersant, which would more strongly repel the non-polar base oil molecules causing more viscous forces in the system.
Figures 18, 19, 20, 21, 22, 23 compare the viscosities of Ultra S4 base oil modified with HMW PIB, HMW uncapped and HMW capped dispersants at 1, 2 and 5 wt % addition levels. At 1 wt % addition level, the addition of either HMW PIB or HMW capped dispersant causes an increase at the 298 K isotherm, but only the addition of HMW capped dispersant causes a noticeable increase in viscosity at all of the isotherms. However, at higher temperatures, the viscosity of Ultra S4 base oil with the addition of PIB, or the uncapped or capped dispersant becomes more similar. The addition of either PIB or HMW capped dispersant causes an increase in viscosity at all isotherms at the 2 wt % addition. At 5 wt %, the addition of any additive causes an increase in viscosity at all isotherms, with the addition of HMW capped dispersant causing the smallest increase in viscosity at 298 K and 323 K isotherms.
It is interesting to note that at 1 wt % addition level, the LMW uncapped dispersant has a greater effect on the system than the HMW uncapped dispersant. It is possible that, as we observed in a previous study [14], the lower molecular weight molecule is able to more easily squeeze in-between molecules in a way that would disrupt the formation of a more regular packing. Also, one must consider that the HMW molecule is about 2.4 times larger than the LMW molecule. Therefore, with a 1 wt % addition, there are 2.4 times more molecules of LMW uncapped dispersant than HMW dispersant, meaning more polar amine heads in the system to create repulsive, viscous forces.
With the HMW additions, the HMW capped dispersant and PIB additives lead to a much higher increase in viscosity of the base oil than their LMW counterparts. It is possible that with less polarity, the increase in viscosity is more dependent on the presence of larger molecules. Meanwhile for the polar molecule, the increase in viscosity might be more dependent on the number of polar heads in the system.
Viscosity data was correlated with density in accordance with the Allal et al. [18] form of the free-volume description of viscosity given by Eq. 5. Figure 24 shows the correlations (as black solid curves) of the viscosity for the base oil modified with LMW uncapped dispersant at 1, 2, and 5 wt % addition levels. The parameters of the viscosity correlations for all the oils are given in Tables 4 and 5. The trends in these parameters are presented in the bar plots shown in Figs. 25, 26, 27.
The addition of either LMW dispersant or LMW PIB causes an increase in the characteristic dissipative length (l). The greatest increase in l occurs with the 1 wt % addition of LMW uncapped dispersant. Although the increase isn’t as drastic with the addition of LMW capped dispersant, this dispersant addition also leads to a consistent increase in the dissipative length. All LMW addition levels of either dispersant or the PIB also lead to a decrease in the diffusive energy barrier (α) and an increase in the free-volume overlap (C).
The increase in the free-volume overlap (C) parameter is also quite high with the addition of 1 wt % LMW capped dispersant. As has been found in a previous study [14], at times a 1 wt % addition has a tendency to disrupt a uniform base oil packing and create more strenuous intermolecular forces in-between molecules. At higher wt % additions, the systems can find a different packing conformation that is more regular and does not lead to as high intermolecular interactions. This might explain why the free-volume overlap parameter can be seen to be higher with the 1 wt % addition of uncapped dispersant. If the molecules undergo higher intermolecular interactions, for example from higher entanglements or greater repulsive forces, an increase in the characteristic dissipation length will occur, as it will naturally take a shorter time for a unit of energy to dissipate, and bf, the dissipation length of a unit of energy, will become smaller. The parameter bf will also naturally become smaller as the free-volume overlap parameter increases, and closer packing will lead to bf dissipating over a shorter length.
The addition of the HMW capped dispersant causes an increase to the characteristic dissipative length (l) at all addition levels. For the HMW capped dispersant or HMW PIB, the 1 wt % addition leads to the smallest value of characteristic dissipative length and the largest value of the diffusive energy barrier (α) compared to the 2 and 5 wt % addition levels. The free-volume overlap (C) is greatest with the 2 and 5 wt % addition of HMW capped dispersant. An interesting note is that with the addition of either additive without a capping group (HMW uncapped dispersant or PIB), the values for the characteristic dissipative length (l) and free-volume overlap (C) are smaller with the 2 wt % addition than at either 1 or 5 wt % addition.
Viscosity data has also been correlated with density using the density-scaling relationship given in Eq. 6. Figure 28 shows the correlation results for Ultra S4 modified with LMW uncapped dispersant. Similar correlations were carried out for all the oil samples. In each system, this scaling procedure unifies the viscosity data at different temperatures and pressures and collapses them into a single curve. Tables 6 and 7 show the correlation parameters for all the oils.
Figures 29, 30, 31 are the bar plots showing how these parameters change with the type of additive and the addition levels. As shown, the A parameter increases in oils modified with LMW uncapped or capped dispersants as well as the LMW PIB. A decrease in parameter B is observed with any addition as well. An increase in the density-scaling parameter γ generally occurs with the addition of either dispersant. However, an increase is not observed for the PIB at the 1 wt % addition level. It is not surprising that with the higher repulsive forces associated with the LMW uncapped dispersants, their addition to the base oil leads to the highest A and γ parameters.
Figures 29, 30, 31 further show that HMW additives display less variation, although the addition of HMW capped dispersants cause an increase in the A parameter. Only the addition of HMW capped dispersant causes a consistent increase in the parameter γ at all addition levels, suggesting that the addition of HMW capped dispersant leads to greater overall repulsive forces. The addition of either HMW uncapped dispersant or PIB causes the A parameter to decrease at 1 wt % addition and increase at 2 wt % addition.
Figures 32, 33, 34 show that all of the density-scaling curves for the oil modified with LMW additives basically all overlap at any addition level, except for the LMW uncapped dispersant at 1 and 5 wt % addition.
Figures 35, 36, 37 show that at the 1 wt % addition level, the HMW uncapped dispersant and PIB cause the largest shift from the base oil. At 2 and 5 wt % addition levels, the scaling curves for the oil modified with HMW uncapped dispersant and PIB are closer to the original base oil than the oil modified with HMW capped dispersant.
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