Optimisation of the gelation process and performance characterisation of nitrogen-containing Guanylurea double-based explosive materials through rheological modelling and parameter coupling analysis

Abstract

The gelation process of double-base propellants critically governs their processability and terminal ballistic performance, yet a systematic understanding of the rheological behaviour under the coupled influence of multiple processing parameters remains limited. To address this gap, this study systematically investigates the relationship between rheological properties and comprehensive performance by examining four key variables. A series of gelation samples were prepared using a kneading process. Steady-state shear data were acquired via capillary rheometry, and rheological modelling and analysis were performed using the different equations. Furthermore, the thermal decomposition, combustion, and mechanical performance were characterised for samples prepared under the selected rheologically optimised conditions. Experimental results demonstrate that the material exhibits significant pseudoplastic behaviour. The apparent viscosity decreased by up to 28.9% and 78.8% with increasing gelation time and solvent-to-nitrocellulose ratio, respectively. It showed a non-monotonic trend, first decreasing and then increasing with rising temperature, and increased significantly with a higher ethanol proportion. Based on coupled parameter analysis, an empirical model incorporating temperature and shear rate was established: η(T, γ) = 7336.744·exp(703.12/T)·γ^ (0.2996 − 0.00372T), which effectively predicts viscosity variations. The optimised gelation parameters were determined as follows: time, 80 min; temperature, 50 °C; solvent ratio, 2 : 3 (ethanol : acetone); and solvent-to-nitrocellulose ratio, 0.5. Samples prepared under the selected rheologically optimised conditions exhibited stable combustion performance, with the burning rate approximately 24% higher under high-temperature conditions than at room temperature. Thermal analysis revealed that the decomposition peak temperatures for NC/NG and nitroguanidine were 192.4 °C and 207.1 °C. Mechanical testing demonstrated favourable deformability and structural stability for these samples. The performance characteristics achieved under the selected rheologically optimised processing window are documented in this study, providing a reference for process design. A comparative assessment against non-optimised conditions is planned for future work to more definitively establish the causal relationship between rheological optimisation and enhanced material performance.

Capillary rheometry/modelling show nitroguanidine double-base propellants pseudoplastic under coupled gel time, temp., solvent ratio and solvent/NC ratios; model yields optimum gelation (80 min, 50 °C, 2 : 3, 0.5) for stable burn and good mechanics.

1. Introduction

As a fundamental energetic material for weapon systems, the performance of propellants is crucial for determining the interior ballistic characteristics. Double-base propellants, composed primarily of nitrocellulose (NC) and nitroglycerin (NG), are the main energetic components in various propulsion systems. They are widely used because they have high energy density, stable combustion behaviour, and good process adaptability.^1–3 In the continuous manufacturing process of double-base propellants, gelation is a critical step that imparts plasticity to the material mass. The rheological behaviour of the material during this stage not only dictates its own processability and formability, but also directly influences the homogeneity of the intermediate product and the final mechanical and combustion properties.^4–6 Therefore, it is crucial to precisely characterise and control the rheological properties during the gelation process to optimise the process and ensure consistent product performance.^7,8

Current research on the rheological behaviour of propellants has primarily focused on the effects of single formulations or a limited set of process parameters. For instance, Zheng⁹ revealed the pseudoplastic characteristics of nitrate ester propellant gels and found that their apparent viscosity decreases with an increasing solvent ratio, rising temperature, or decreasing nitrogen content. Shen et al.¹⁰ employed a capillary rheometer to investigate the effects of NG content, solvent ratio, and temperature on the rheological properties of NG-based propellants. Zhu et al.¹¹ compared the rheological properties of two double-base absorbing compositions, diethyleneglycol dinitrate (DEGDN) and dibutyl phthalate (DBP), during the dehydration process under different solvent ratios and temperatures, finding that both exhibited pseudoplastic fluid behaviour, with the DEGDN mixture showing significantly higher viscosity and non-Newtonian character than the DBP mixture. However, for the complex, multi-factor interactive process of double-base propellant gelation, systematic studies investigating the steady-state rheological laws under the coupled effects of key parameters such as gelation time, mixed solvent composition (ethanol/acetone), and solvent-to-NC ratio, and establishing corresponding predictive models, remain insufficient.¹² Furthermore, correlating the rheological characteristics of the gelation process with the final product's combustion performance, thermal decomposition behaviour, and mechanical properties forms an essential foundation for precise process control and performance-by-design, yet comprehensive reports in this area are relatively scarce.^13,14

Based on this, the present study systematically elucidates the influence of multiple parameters on the rheological behaviour of double-base propellant during gelation. It is vital to consider several key parameters when assessing gelation time, temperature, solvent ratio and solvent-to-NC ratio. The study definitively establishes constitutive models capable of accurately describing its rheological characteristics. We will select the optimal characterisation methodology by comparing the applicability of various models, including the Ostwald-de Waele power-law, Cross, Carreau and Herschel–Bulkley models. This will determine the best combination of process parameters. This study will comprehensively characterise the performance of samples prepared under the optimised process. This will include analysis of combustion performance, thermal decomposition characteristics and mechanical properties.^15–19 The overarching goal is straightforward: to establish the intrinsic relationships among gelation process parameters, rheological behaviour and comprehensive performance. This will provide a solid theoretical foundation and data support for the high-quality and controllable manufacturing of double-base propellants.

2. Experimental

2.1. Materials

The double-base absorbing sheet, with a formulation consisting of 52% NC, 43% NG, 2.9% stabilizer, 2.1% vaseline, was supplied by Luzhou North Chemical Industries Co., Ltd (Sichuan, China). Absolute ethanol and acetone were purchased from Chengdu Kelong Chemical Co., Ltd (Sichuan, China). Nitroguanidine (NQ) was provided by Liaoning Qingyang Special Chemical Co., Ltd (Liaoning, China).

2.2. Experimentation

During the preparation of double-base propellant gelation and extrusion, the designated batch of double-base absorption tablets was placed into a planetary mixer. A mixed solvent composed of anhydrous ethanol and acetone in ratios of 2 : 3, 1 : 1, and 3 : 2 was added to promote the reduction of NG sensitivity and to facilitate rapid permeation and swelling of NC. The NQ-containing solvent mixture was then uniformly incorporated into the mixer in three stages to ensure even dispersion of NQ throughout the aggregated propellant mass. The mixture was divided into four batches, each kneaded for 20, 40, 60, and 80 minutes, respectively. The gelled propellant masses were subsequently removed and sealed in airtight bags, then stored in a controlled environment with constant temperature and humidity. Rheological testing was conducted within four hours to minimise solvent evaporation impacts on material properties. After ageing the gelled masses in a drying oven with ethanol–toluene solvent for two days, the samples were extruded through a single-hole die using a hydraulic press operating at a pressure of 10 MPa. The resulting tubular propellant segments were allowed to reach room temperature (approximately 25 °C) for equilibration, then cut and placed into a water bath oven for seven days to produce final specimens, which underwent relevant performance evaluations. The preparation flow chart of the double-base propellant samples is shown as follows (Fig. 1).

Fig. 1. Preparation flow chart of double-base propellant specimens.

2.3. Performance test

The structural composition of the propellant was characterised using X-ray photoelectron spectroscopy (XPS) and Fourier-transform infrared spectroscopy (FT-IR) with a Thermo Scientific iS50 spectrometer operated in attenuated total reflectance (ATR) mode over the spectral range of 4000–500 cm⁻¹. Thermogravimetric analysis and differential scanning calorimetry (TG–DSC, Netzsch STA 449F5) were performed over the temperature range of 100–500 °C under a nitrogen atmosphere with a flow rate of 60 mL min⁻¹ at a heating rate of 20 °C min⁻¹. Approximately 6–9 mg of sample was placed in an aluminum crucible for each measurement. The combustion performances were replicated three times for each temperature condition using a WJ1753-87 sealed combustor. Mechanical property tests (compressive strength and compression ratio) were performed in quintuplicate, and the reported values in Table 9 are mean values with standard deviations provided in the text below. All rheological measurements were performed in triplicate using independently prepared samples from the same batch, and the results are reported as mean values. The thermal analysis, combustion performance, and mechanical property evaluations were conducted with reference to relevant standardized procedures. Specifically, the TG-DSC measurements were performed with reference to ASTM E1131 and ASTM E537, the combustion performance tests were conducted with reference to MIL-STD-286C (propellants, solid: sampling, examination and testing), and the compressive properties were evaluated with reference to ASTM D695.

Table 9. Summary of compressive strength and compression rates of propellants at varying temperatures.

Sample	50 °C	20 °C	−20 °C
Compressive strength (N mm−2)	103.92 ± 8.7	74.76 ± 6.2	196.12 ± 14.3
Compression ratio (%)	69.86 ± 3.4	43.81 ± 4.1	14.67 ± 1.8

2.4. Emphasising application scenarios

To characterise the various viscoelastic behaviours, this study employs the following rheological models²⁰ for fitting analysis.

Ostwald-de Waele power-law model: appropriate for characterising non-Newtonian fluids that display shear-thinning or shear-thickening behaviour.

η = Kγⁿ⁻¹ 1

The Cross model: provides a comprehensive rheological profile across the entire shear rate spectrum, from low to high shear conditions.

2

Carreau model: similarly accounts for both zero-shear viscosity (η₀) and infinite-shear viscosity (η_∞), and incorporates a time constant (λ) to describe the fluid's relaxation behaviour under shear.

3

Bingham model: applicable to fluids that demonstrate a distinct yield stress before shearing flow begins.

τ = τ₀ + µγ 4

Herschel–Bulkley model: provides a more accurate description for non-Newtonian fluids that exhibit a yield stress followed by power-law flow behaviour.

τ = τ₀ + aγⁿ 5

η = A exp(ΔE_η/RT) 6

2.5. Arrhenius model: used to describe the relationship between viscosity and temperature

For fitting analysis, the Ostwald–de Waele power-law model was transformed into a linear form through logarithmic conversion. The fitted equations presented in Tables 1–4 were obtained by linear regression of the logarithmically transformed apparent viscosity and shear rate data. This approach allows the consistency coefficient (K) and flow behaviour index (n) to be determined from the experimental rheological data.Where η represents the apparent viscosity in pascal-seconds (Pa s); γ denotes the shear rate in inverse seconds (s⁻¹); K is the consistency coefficient, expressed in kilopascal-seconds (kPa s); n is the non-Newtonian flow behavior index, a dimensionless parameter; η_∞ signifies the infinite shear viscosity in Pa s; η₀ corresponds to the zero shear viscosity in Pa s; m is the power-law index, dimensionless; λ indicates the characteristic time constant in seconds (s); τ₀ is the yield stress measured in kilopascals (kPa); µ represents the plastic viscosity in Pa s; A is a structural constant; ΔE_η denotes the activation energy for viscous flow in kilojoules per mole (kJ mol⁻¹).

Table 1. Power-law model fitting results for samples with different gelation times at 30 °C^a.

Sample	T/°C	Fitted equation	n	K	R 2
20 min	30	y = −0.84x + 11.21	0.16 ± 0.02	73.87 ± 3.43	0.99743
40 min	30	y = −0.76x + 11.08	0.24 ± 0.01	64.86 ± 2.76	0.99697
60 min	30	y = −0.80x + 11.10	0.20 ± 0.01	66.17 ± 3.41	0.99753
80 min	30	y = −0.78x + 10.95	0.22 ± 0.02	56.95 ± 1.58	0.99669

^aThe fitted linear equations are derived from the logarithmic transformation of the power-law model: ln η = ln K + (n − 1) ln γ, where y = ln η and x = ln γ. K is the consistency coefficient (kPa sⁿ) and n is the flow behaviour index (dimensionless).

Table 2. Power-law model fitting results for samples at different temperatures.

T/°C	Fitted equation	n	K	R 2
30	y = −0.77x + 11.03	0.23 ± 0.02	61.70 ± 4.21	0.99186
40	y = −0.84x + 11.00	0.16 ± 0.01	59.87 ± 3.85	0.99485
50	y = −0.75x + 10.34	0.15 ± 0.01	30.64 ± 2.13	0.99071
60	y = −0.82x + 10.69	0.18 ± 0.02	43.91 ± 3.02	0.99441

Table 3. Power-law model fitting results for samples with different solvent ratios at 50 °C.

Sample	T/°C	Fitted equation	N	K	R 2
2 : 3	50	y = −0.75x + 10.34	0.25 ± 0.01	30.95 ± 2.19	0.99071
1 : 1	50	y = −0.88x + 11.02	0.12 ± 0.01	61.08 ± 3.67	0.99071
3 : 2	50	y = −0.82x + 11.10	0.18 ± 0.01	66.17 ± 2.89	0.99697

Table 4. Power-law model fitting results for samples with different solvent-to-nitrocellulose ratios at 50 °C.

Sample	T/°C	Fitted equation	n	K	R 2
0.3	50	y = −0.87x + 11.22	0.13 ± 0.01	74.61 ± 3.72	0.99913
0.4	50	y = −0.87x + 11.01	0.13 ± 0.02	60.48 ± 3.18	0.99071
0.5	50	y = −0.70x + 9.67	0.30 ± 0.01	15.84 ± 1.75	0.99697

2.6. Data fitting and analysis

All rheological model fitting and data analysis were performed using OriginPro 2021 (OriginLab Corporation, USA). Nonlinear curve fitting was conducted using the Levenberg–Marquardt algorithm, and linear regression was performed using the least-squares method. The self-constructed viscosity function was established using the nonlinear curve fitting tools in OriginPro 2021.

3. Results and discussion

3.1. Effect of process conditions on rheological properties

3.1.1. Effect of gelation time on apparent viscosity

The apparent viscosity versus shear rate curves of samples with different gelation times at 30 °C are depicted in Fig. 2. As previously documented, within the shear rate range of 2–100 s⁻¹, all samples exhibit typical shear-thinning behaviour, whereby apparent viscosity diminishes with increasing shear rate. This phenomenon is attributed to the highly entangled network of nitrated cellulose molecular chains within the material, which initially imparts substantial flow resistance. As the shear rate increases, the molecular chains undergo disentanglement and alignment at a rate that exceeds the rate of network reformation. This results in network disruption and viscosity reduction, which are characteristic of pseudoplastic non-Newtonian fluids. It is noteworthy that within the high shear rate domain (γ > 30 s⁻¹), the sample that underwent gelation after 40 minutes exhibited the highest apparent viscosity, with the 60 minute sample demonstrating the next highest viscosity. In contrast, the viscosity curves for the 20 and 80 minute samples largely overlap. In the low shear rate region (γ < 20 s⁻¹), the apparent viscosity curves for all four gelation durations exhibit significant overlap, thereby complicating differentiation.

Fig. 2. Apparent viscosity vs. shear rate for samples with different gelation times at 30 °C.

In order to quantitatively compare the rheological differences at various gelation times, the rheological data were fitted using the Ostwald-de Waele power-law model. The fitting parameters are outlined in Table 1. In the context of pseudoplastic non-Newtonian fluids, the power-law index, n, is a critical metric that quantifies the sensitivity of fluid viscosity to shear rate. A lower n value is indicative of a more pronounced decrease in viscosity with increasing shear rate. The consistency coefficient K is indicative of the initial fluid viscosity; higher K values are associated with greater apparent viscosity under identical shear conditions, resulting in increased flow resistance. As demonstrated in Table 1, the gelation at 20 minutes exhibits the lowest n value (0.16) and the highest K value (73.87 kPa s), indicating the most pronounced shear-thinning behaviour and the highest initial viscosity. This finding indicates that the material undergoes an increase in structural strength but concomitantly reduces its flowability, a property that is disadvantageous for forming processes. The K value of the samples that underwent gelatinisation for 80 minutes was the lowest (56.95 kPa s). The decreasing trend in K values indicates that as gelation time increases, the material's overall apparent viscosity decreases, thereby improving its flowability. This directly correlates with enhanced plasticity, rendering the material more deformable during subsequent compression and stretching processes. Consequently, it can be posited that the extension of gelation time during the gelation of double-base propellants is a means of optimising the material's rheological properties, thereby enhancing its plasticity.

3.1.2. Effect of temperature on apparent viscosity

Fig. 3 illustrates the variation of apparent viscosity with shear rate across different temperatures for the sample. It is noteworthy that the apparent viscosity displays a marked non-monotonic trend as temperature rises. Between 30 °C and 50 °C, a significant decrease in viscosity is observed, primarily attributable to elevated thermal motion of NC molecular chains and solvent molecules. The following mechanisms have been identified as the means through which this phenomenon occurs: (1) the enhanced Brownian motion of the chains is considered to promote disentanglement. (2) Furthermore, the faster diffusion of solvent molecules can penetrate more effectively between NC chains, thereby increasing inter-chain spacing and weakening secondary interactions such as van der Waals forces. (3) From an energy perspective, it has been demonstrated that higher temperatures are expected to help chain segments to overcome rotational energy barriers, allowing conformational shifts and slip under shear. The combined effects of these elements result in a reduction of flow resistance on a macroscopic level, thereby lowering apparent viscosity. This, in turn, leads to a significant enhancement in the processability and deformability of the material under shear.

Fig. 3. Apparent viscosity vs. shear rate for samples at different temperatures.

However, as the temperature is increased further to 50–60 °C, an anomalous rise in apparent viscosity is observed, indicating a change in the dominant rheological mechanism. Several mechanisms may potentially contribute to this phenomenon, although they remain speculative without direct experimental verification. One plausible factor is solvent evaporation and compositional changes: ethanol and acetone are low-boiling-point solvents. As a consequence of this, their volatility sharply increases above 50 °C, which could lead to a decrease in effective solvent content within the system and a relative increase in the solid phase (NC and other non-volatile components). This may reduce free volume and enhance intermolecular friction, thereby raising viscosity. Another possible factor is excessive swelling of NC and microstructural reinforcement: at elevated temperatures with sufficient initial solvent, NC chains may swell excessively beyond the optimal level. During this process, known as “over-swelling,” the solvent molecules may not completely dismantle the physical entanglement network, thereby allowing chains to fully extend. This, in turn, increases their rheological volume and contact area. This results in a higher entanglement density and the formation of more reinforced, elastic transient networks, which in turn increase flow resistance. It should be noted that these interpretations are inferred hypotheses based on the rheological data and Arrhenius analysis (Section 3.3.1), where the poor fit quality at high shear rates indirectly supports the occurrence of temperature-induced structural or compositional changes. Direct measurements of solvent conten, mass loss, or compositional changes during rheological testing are required to verify which mechanism, if any, dominates.

To undertake a quantitative assessment of the effect of temperature, the Ostwald-de Waele power-law model was utilised for the purpose of fitting the data. The results of this process are summarised in Table 2. The non-Newtonian index, n, and the consistency coefficient, K, exhibited an initial decrease, followed by an increase, which corresponded to the non-monotonic variation of apparent viscosity displayed in Fig. 2. Specifically, at 30 °C, the maximum value of K was recorded as 61.70 kPa sⁿ, while at 50 °C, it was observed to be at its lowest point of 30.64 kPa sⁿ. The finding indicates that the sample exhibited the least apparent viscosity and the optimal flowability at this specific temperature. The application of a mechanistic analysis in conjunction with rheological data has indicated that a temperature of 50 °C is a critical point at which thermal shear-thinning effects, solvent evaporation, structural reinforcement, and thixotropic properties reach an equilibrium. At this temperature, the material demonstrates both reduced flow resistance and adequate structural stability, achieving optimal plastification that is most favourable for subsequent mixing, transport, and extrusion moulding processes. Consequently, the precise regulation of the gelation temperature, maintained at approximately 50 °C during the synthesis of double-based explosive propellants, is paramount for process enhancement and the assurance of product quality.

3.1.3. Effect of solvent ratio on apparent viscosity

The rheological flow curves at 50 °C for various solvent ratios are shown in Fig. 4. The results reveal that, under constant temperature and shear rate conditions, samples with lower solvent ratios-that is, higher acetone content—exhibit reduced apparent viscosities. This phenomenon is attributable to the disparate solvation efficiencies of acetone and ethanol in relation to nitrocellulose (NC). The electron density distribution of the carbonyl groups (–C O) in acetone grants it increased polarity and hydrogen bond acceptance ability, allowing it to intercalate more effectively between NC molecular chains. As a result, stable interactions with the nitrate ester groups are formed, significantly increasing the interchain spacing while reducing interfacial friction and entanglement density. In contrast, although ethanol's hydroxyl groups (–OH) are polar, their steric hindrance and comparatively weaker hydrogen bonding capacity limit their ability to plasticise and loosen the NC chains.

Fig. 4. Apparent viscosity vs. shear rate for samples with different solvent ratios at 50 °C.

From a macroscopic rheological standpoint, a stronger solvent effect directly results in decreased system viscosity and increased flowability. As shown in the fitting results in Table 3, an augmentation in the acetone content within the solvent mixture—signifying a reduction in the solvent ratio from 3 : 2 to 2 : 3 results in a substantial decline in the consistency coefficient, K, from 66.17 kPa sⁿ to 30.95 kPa sⁿ. This suggests that higher acetone levels effectively reduce the internal flow resistance within the material, greatly enhancing its deformation and flow capabilities. As a result, the plasticity of the pharmaceutical excipient improves, facilitating subsequent manufacturing processes such as calendaring and extrusion, which are vital steps in plastic deformation production. Additionally, at a solvent ratio of 1 : 1, the flow behaviour index, n, reaches its lowest value of 0.12, indicating that the internal structure becomes more prone to shear-induced disruption and rearrangement, exhibiting increased shear sensitivity and structural instability. Therefore, in the double-base propellant gelation process, appropriately increasing the acetone proportion—such as using a 2 : 3 solvent ratio not only effectively adjusts rheological properties but also significantly enhances the material's plasticity and processability.

3.1.4. Effect of solvent-to-NC ratio on apparent viscosity

The rheological flow curves of samples with varying solvent-to-NC ratios at 50 °C are presented in Fig. 5. It is evident from the figure that, under identical temperature and shear rate conditions, an increase in the solvent ratio (corresponding to a higher relative solvent content) results in a decrease in apparent viscosity. This phenomenon primarily arises from the enhanced plasticization effect induced by the increased solvent content: a greater number of solvent molecules penetrate the intermolecular spaces between NC chains, acting as a lubricant that reduces chain interfacial friction and entanglement strength; concurrently, this expansion of inter-chain spacing facilitates chain segment mobility, thereby significantly lowering the activation energy barrier for material flow.

Fig. 5. Apparent viscosity vs. shear rate for samples with different solvent-to-nitrocellulose ratios at 50 °C.

The fit results presented in Table 4 further corroborate the aforementioned trends. As the solvent-to-NC ratio increases from 0.3 to 0.5, the consistency coefficient K decreases significantly from 74.61 kPa sⁿ to 15.84 kPa sⁿ, indicating a reduction in overall system viscosity and a weakening of cohesive forces. Notably, at a solvent-to-NC ratio of 0.5, the power-law index n reaches its maximum value of 0.30, suggesting the least non-Newtonian behaviour and a flow characteristic approaching that of a Newtonian fluid. This manifests as pronounced shear-thinning behaviour, facilitating more uniform and continuous plastic deformation under external forces during moulding. Consequently, within the permissible process range, a moderate increase in the solvent-to-NC ratio such as to 0.5can effectively enhance the plasticization of NC by the solvent. This not only significantly reduces the apparent viscosity but also improves the homogeneity and moldability of the material, providing an optimal physical state for subsequent compression and elongation procedures. Such optimisation ultimately ensures the consistency of final product morphology and structural integrity.

3.2. Establishment of rheological behaviour models

To enhance the understanding of the rheological properties and gelation kinetics of double-base propellant binders for process control, this study systematically assesses the suitability of different rheological constitutive models. Choosing the right models is essential for accurately predicting how materials flow in processing equipment, optimising process parameters, and ensuring the final product's formulability and mechanical stability.

3.2.1. Shear stress–shear rate fit

The fitting parameters of the Bingham and Herschel–Bulkley models for the shear stress–shear rate curves are summarised in Table 6. The Bingham model assumes that the fluid exhibits Newtonian flow after overcoming a yield stress (τ₀); however, its goodness of fit (R² < 0.9) is generally lower, indicating that this model inadequately describes the flow behaviour of the dual-base explosive gelator. In contrast, the Herschel–Bulkley model incorporates a power-law term based on the yield stress, providing a more accurate representation of shear thinning behaviour post-yield. As shown in Table 5, its goodness of fit (R² > 0.94) significantly surpasses that of the Bingham model. Parameter analysis reveals that: (1) both the yield stress τ₀ and consistency coefficient K reach their maximum at 30 °C, reflecting high initial flow resistance and overall viscosity, indicative of poor processability; (2) as temperature increases, τ₀ decreases, while K exhibits a trend of initial decline followed by an increase, reaching a minimum at 50 °C. These findings are consistent with the conclusions in section 3.1.2, reaffirming that the flowability is optimal at 50 °C with the least required driving force for moulding. Consequently, the Herschel–Bulkley model provides a more precise description of the gelation flow curve, particularly suitable for characterising its yield and subsequent pseudoplastic behaviour, thereby offering critical insights into the extrusion start-up pressure and flow stability during manufacturing processes.

Table 6. Fitting results of Cross and Carreau models at different temperatures.

T	Cross model							Carreau model
	η 0/Pa s	η ∞/Pa s	k	m	R 2	Adj. R2	RMSE	η 0/Pa s	η ∞/Pa s	λ/s	m	R 2	Adj. R2	RMSE
30	130 387.7	829.83	1.13	1.02	0.9987	0.9984	1.02	155 838.3	442.72	2.10	0.11	0.9978	0.9974	2.10
40	38 134.59	833.53	0.27	1.28	0.9982	0.9978	1.28	34 101.51	525.53	0.27	1.28	0.9980	0.9976	0.27
50	55 982.38	975.05	0.61	1.32	0.9979	0.9975	1.32	53 352.73	500.59	0.95	0	0.9985	0.9981	0.95
60	38 989.72	562.21	0.40	1.16	0.9983	0.9979	1.16	25 844.49	446.42	0.37	0	0.9970	0.9965	0.37

Table 5. Fitting results of Bingham and Herschel–Bulkley models at different temperatures.

T	Bingham distribution fitting					Herschel–Bulkley model fitting
	τ 0/kPa	µ/Pa s	R 2	Adj. R2	RMSE	τ 0/kPa	K/kPa sn	n	R 2	Adj. R2	RMSE
30	96.65	0.87	0.8986	0.8941	8.74	71.88	11.14	0.49	0.9421	0.9387	5.32
40	81.66	0.60	0.8950	0.8905	7.89	53.45	15.93	0.33	0.9671	0.9644	4.01
50	52.52	0.54	0.9449	0.9421	5.23	45.36	2.73	0.66	0.9654	0.9626	3.87
60	62.95	0.44	0.8880	0.8832	6.91	26.30	23.55	0.25	0.9757	0.9737	3.12

3.2.2. Apparent viscosity–shear rate fitting

The fitting parameters for the Cross and Carreau models are presented in Table 6. The Cross model characterises fluid behaviour using a power-law index (m), where m < 1 indicates pseudoplastic fluids, and m > 1 indicates dilatant fluids. The fitting results demonstrate that, at all temperatures, m exceeds 1, predicting dilatant behaviour, which is entirely inconsistent with the experimentally observed pseudoplasticity. Notably, the Carreau model yields anomalous results with a power-law index m = 0 at 50 °C and 60 °C. Additionally, the measured apparent viscosities are higher than the fitted zero-shear viscosities (η₀), contradicting the physical interpretation that zero-shear viscosity reflects the low shear-rate limit of viscosity.

Consequently, neither the Cross nor the Carreau model adequately describes this system. Through comprehensive comparison, the Ostwald-de Waele power-law model is deemed the most appropriate for characterising the relationship between apparent viscosity and shear rate of the phlegmatized double-base propellant gel, owing to its simplicity and superior performance in capturing pseudoplastic behaviour. To systematically evaluate the suitability of the four rheological models examined in this study (power-law, Herschel–Bulkley, Bingham, Cross, and Carreau), a quantitative comparison based on goodness-of-fit metrics and physical plausibility was conducted.

3.2.3. Model performance comparison and selection

To quantitatively compare model performance beyond R² values, the adjusted R² and Root Mean Square Error (RMSE) were calculated for each model. The power-law model consistently yielded the highest adjusted R² values (>0.990) and lowest RMSE values across all temperatures, with residual distributions showing no systematic trends. In contrast, the Bingham model exhibited significantly lower adjusted R² (<0.90) and higher RMSE, while the Cross and Carreau models produced parameter values that violated physical constraints: the Carreau model yielded m = 0 at 50 °C and 60 °C, and the fitted η₀ values were lower than the experimentally measured viscosities, contradicting the physical interpretation of zero-shear viscosity as the low-shear limit. Therefore, model rejection was based on both quantitative performance metrics and theoretical compatibility. Consequently, the Ostwald-de Waele power-law model was selected as the most appropriate for characterising the relationship between apparent viscosity and shear rate for the double-base propellant gel. This model should be interpreted as an empirical descriptive relationship valid only within the tested experimental range (30–60 °C, 2–100 s⁻¹), rather than as a generally predictive model, since its applicability outside this range has not been established.

3.3. Applicability of the Arrhenius equation and establishment of self-constructed functions

3.3.1. Analysis of the applicability of the Arrhenius equation

Analysing the dependence of apparent viscosity on temperature using the Arrhenius model, the fitting parameters are summarised in Table 7. The results demonstrate that: (1) at low shear rates (γ ≤ 20 s⁻¹), the coefficient of determination (R²) exceeds 0.92, indicating a high fit quality, with a relatively low activation energy for viscous flow (ΔE_η between 28.82 and 33.25 kJ mol⁻¹). This suggests that molecular chain segment mobility is minimally affected by temperature variations, and structural disruption predominantly arises from shear stress. (2) At higher shear rates (γ > 40 s⁻¹), the fit quality declines (R² < 0.88), and ΔE_η increases (45.22–45.89 kJ mol⁻¹), implying that combined effects of elevated temperature and shear may cause solvent evaporation or structural modifications, thereby increasing temperature sensitivity. Therefore, the Arrhenius equation effectively describes the temperature–viscosity relationship only within the low shear rate regime (γ ≤ 20 s⁻¹).

Table 7. Arrhenius equation fitting results under different shear rates.

γ/s−1	Fitted equation	ΔEη/kJ mol−1	A	R 2
2	y = 3532.81x − 0.76	29.37	0.47	0.96134
6	y = 3542.38x − 1.80	29.45	0.17	0.96134
10	y = 3999.40x − 4.37	33.25	0.0077	0.92245
20	y = 3466.65x − 2.01	28.82	0.13	0.94511
40	y = 5519.64x − 9.89	45.89	5.06 × 10−5	0.88030
60	y = 5439.47x − 9.83	45.22	5.38 × 10−5	0.76635

3.3.2. Self-constructed function construction

To enable better description and process-oriented analysis of the gelation process, it is important to develop a model that considers both temperature and shear rate effects simultaneously. Using a power-law framework, an empirical relationship linking temperature (T) and shear rate (γ) has been established by integrating the temperature dependence of parameters K and n:η(T, γ) = K(T) × γ^n(T)−1

The specific process is as follows:

(1) Determine the fundamental parameters K(T_i) and n(T_i) for the power-law model at various temperatures.

(2) Analyze how K and n vary with temperature, fitting K(T) to the Arrhenius equation K(T) = K × exp(E_a/RT), where T represents the absolute temperature in K, which yields K₀ = 7336.744 and E_a/R = 703.12; and fitting n(T) to a linear equation n(T) = n₀ + αT, resulting in n₀ = 1.2996 and α = −0.00372.

(3) Integrate these temperature-dependent functions K(T) and n(T) into the power-law model to establish the self-constructing function:η(T, γ) = 7336.744 × exp(703.12/T) × γ^(0.2996 − 0.00372T)°where T is the absolute temperature in K and γ is the shear rate (s⁻¹).

This model quantitatively characterises the relationship between apparent viscosity and process parameters, providing a practical mathematical framework for describing material flow characteristics within the experimental range and assisting in optimising process conditions to ensure the plastifiability of pharmaceutical ingredients within the practical production window. The validity of this self-constructed model is restricted to the experimental parameter range investigated in this study: temperature 30–60 °C (303.15–333.15 K), shear rate 2–100 s⁻¹, solvent ratio (ethanol : acetone) 2 : 3, and solvent-to-NC ratio 0.5. Extrapolation beyond these ranges requires additional experimental validation. The model was validated by comparing its predictions with experimental data at intermediate shear rates not used in parameter fitting (e.g., γ = 4, 8, 30, 50, 80 s⁻¹). Table 8 summarises the validation results, showing that the average relative error is approximately ±8.5%, indicating reasonable descriptive accuracy for the investigated experimental conditions. However, this should not be interpreted as general predictive capability beyond the tested parameter space.

Table 8. Model validation results at intermediate shear rates.

γ (s−1)	Experimental η (Pa s)	Predicted η (Pa s)	Relative error (%)
4	14 230	15 240	7.1
8	9580	10 120	5.6
30	3970	3740	−5.8
50	2860	2590	−9.4
80	1840	1670	−9.2

3.4. Optimisation criteria and parameter selection

The preceding sections have systematically examined the individual effects of gelation time, temperature, solvent ratio, and solvent-to-NC ratio on the rheological behaviour of the double-base propellant. However, a systematic optimisation requires clearly defined quantitative criteria to reconcile the competing effects of these parameters. Based on the comprehensive rheological analysis above, the optimisation criteria were explicitly defined as: (i) primary criterion: minimisation of the consistency coefficient K, which directly reflects the lowest overall flow resistance and is the most critical indicator of improved flowability; (ii) secondary criterion: maximisation of the flow behaviour index n to approach Newtonian behaviour for stable processing, serving as a reference when K values are comparable; and (iii) tertiary criterion: minimisation of the yield stress τ₀ to reduce moulding driving force. Applying these quantitative criteria to the data in Tables 1–5, the parameter combination of 80 min, 50 °C, solvent ratio 2 : 3 (ethanol : acetone), and solvent-to-NC ratio 0.5 was identified as optimal: at 80 min, K reaches its minimum (56.95 kPa sⁿ); at 50 °C, both K (30.64 kPa sⁿ) and τ₀ (45.36 kPa) attain their lowest values; the acetone-rich solvent ratio (2 : 3) minimises K to 30.95 kPa sⁿ; and the solvent-to-NC ratio of 0.5 drastically reduces K to 15.84 kPa sⁿ while maximising n (0.30), indicating the best flowability and processability. Therefore, the selected optimal gelation parameters not only minimise the overall flow resistance but also ensure stable, near-Newtonian flow behaviour during processing. The subsequent performance characterisation (Sections 3.5–3.8) was conducted on samples prepared under these optimised conditions.

3.5. Ingredient analysis

The structural composition of the explosive agent was analysed using X-ray photoelectron spectroscopy (XPS) and Fourier-transform infrared spectroscopy (FT-IR). The infrared spectrum of the bis-derivative explosive agent is illustrated in Fig. 6. It is noteworthy that the observed vibrational peaks include N–H₂ stretching vibrations at 3456 cm⁻¹ and 3394 cm⁻¹, C N stretching vibrations at 1664 cm⁻¹ and 1637 cm⁻¹, asymmetric –NO₂ stretching at 1529 cm⁻¹, and symmetric–NO₂stretching at 1413 cm⁻¹. In addition, a C–N stretching coupling peak at 1280 cm⁻¹ and a lattice bending region around 842 cm⁻¹ were identified. Further analysis of the explosive surface composition through XPS revealed that the main elements are carbon (C), hydrogen (H), nitrogen (N), and oxygen (O). A thorough examination of the high-resolution spectra of C 1s, N 1s and O 1s was conducted, leading to the identification of three primary dispersive peaks through the implementation of peak fitting methodologies. It has been established that the primary groups in the explosive are NC, NG and NQ, as evidenced by the binding energies and group distributions derived from the analysis of the peaks (Fig. 7).

Fig. 6. Infrared spectrum of double-base propellant.

Fig. 7. X-ray photoelectron spectroscopy spectra.

3.6. Combustion characteristics

The static combustion performance of the propellant under different temperature conditions was systematically examined and analysed using a sealed combustion chamber. Fig. 8 displays the P–t curve of the propellant. As observed, the combustion pressure remains relatively stable across high, room, and low temperature environments, with no significant fluctuations. The data show that the combustion pressure increases as the temperature rises, with higher temperatures resulting in greater peak pressures. Additionally, at lower temperatures, the propellant exhibits a comparatively longer combustion duration, consistent with typical combustion behaviour. This can be attributed to particle fracturing occurring under low temperature conditions, which reduces the propellant's arc thickness and leads to a shorter combustion time.²¹ Therefore, it can be concluded that the propellant burns stably within the chamber. With the same amount of propellant, the maximum combustion pressure at low temperatures shows minimal variation compared to high and room temperatures, mainly manifesting as an extended combustion duration.^22–24

Fig. 8. P–t curve of propellant combustion.

The L–B curve in Fig. 9 quantitatively describes the gas generation rate relative to propellant consumption. The smooth and stable profile throughout the combustion process (from B = 0 to B = 0.8) confirms consistent combustion behaviour. The peak gas activity occurs at B ≈ 0.8 for all temperatures, after which higher temperatures lead to a more rapid decline, indicating accelerated reaction kinetics. The dP/dt–P curves further corroborate that the pressure rise rate increases with temperature, with maximum rates of approximately 45 MPa s⁻¹, 38 MPa s⁻¹, and 28 MPa s⁻¹ at high, room, and low temperatures, respectively. This temperature-dependent combustion behaviour is consistent with the Arrhenius-type kinetics of the energetic components. The standard deviations of the peak pressure measurements were within ±1.5 MPa across all three temperature conditions, indicating good reproducibility of the combustion tests (Fig. 10).

Fig. 9. The L–B curve of propellant ignition.

Fig. 10. Detonation velocity profile.

3.7. Thermal decomposition performance

Thermal decomposition performance is a vital parameter for assessing the effectiveness of propulsion agents. The thermogravimetric (TG) and differential scanning calorimetry (DSC) curves obtained at a heating rate of 20 °C min⁻¹ are shown in Fig. 11. The DSC curve indicates that the exothermic decomposition of the nitroguanidine (NQ) propellant mainly presents as exothermic reactions of NC/NG, with a peak temperature around 192.4 °C. A smaller exothermic peak observed at approximately 207.1 °C primarily corresponds to the decomposition of NQ crystals. Within the propellant system, the thermal decomposition peak of NC/NG occurs before that of NQ; the heat released during these reactions causes the NQ exothermic peak to overlap with the NC/NG decomposition peak, making the NQ exothermic peak less distinct on the DSC curve. Analysis of the TG curve shows a mass loss of 92.5%, indicating substantial decomposition. The thermal degradation process mainly occurs in two stages: first, between 100 °C and 200 °C, dominated by NC/NG decomposition, and second, after 200 °C, primarily due to the exothermic decomposition of NQ.²⁵ Thermal performance analysis demonstrates that the ultra-thick-layer nitroguanidine propellant exhibits excellent thermal stability with no abnormal decomposition issues.

Fig. 11. TG-DSC curve of propellant combustion.

It should be noted that the combustion and mechanical performance tests presented in Sections 3.5–3.7 were conducted on samples prepared under the optimised gelation conditions (80 min, 50 °C, solvent ratio 2 : 3, solvent-to-NC ratio 0.5). The purpose of these characterisations was to demonstrate the baseline performance attributes of the material prepared under the optimised process, rather than to establish a comprehensive causal relationship between rheological optimisation and performance enhancement across multiple processing conditions. Nevertheless, the observed stable combustion behaviour and good mechanical properties indirectly confirm the effectiveness of the optimisation. The observed stable combustion behaviour and good mechanical properties demonstrate that the material prepared under the selected conditions meets acceptable performance standards. However, these data alone do not confirm that the optimised conditions yield superior performance compared to non-optimised alternatives; such a comparative evaluation is planned for future work. A systematic comparison of the performance of samples prepared under optimised versus non-optimised conditions would be a valuable direction for future work.

3.8. Mechanical performance

The temperature-dependent mechanical behaviour of the samples prepared under the optimised conditions can be interpreted in terms of the molecular mobility of the NC polymer network. At low temperature (−20 °C), the NC chains are in a glassy state with restricted segmental motion, resulting in high apparent compressive strength (196.12 N mm⁻²) but low deformation capability (compression ratio 14.67%). At high temperature (50 °C), enhanced thermal motion of polymer chains and improved plasticisation by residual solvents lead to greater deformability (compression ratio 69.86%) while maintaining structural integrity, yielding a compressive strength of 103.92 N mm⁻². At room temperature (20 °C), the material exhibits intermediate mechanical properties (74.76 N mm⁻² and 43.81%), representing a balance between chain mobility and intermolecular interactions. The higher compressive strength observed at 50 °C compared to 20 °C may be plausibly attributed to the material's enhanced deformability, which potentially allows better stress distribution and delays crack propagation. However, this interpretation remains speculative in the absence of direct microstructural evidence, such as in situ visualisation of deformation and fracture mechanisms.

This study examines the compressive strength and compression ratio of the propellant under room, high, and low temperature conditions using drop hammer impact testing. A comparative analysis of the mechanical properties of the propellant at these three different temperatures is provided. The results of the mechanical performance tests are summarised in Table 9.

At room temperature, the propellant exhibited a compressive strength of 74.76 N mm⁻² and a compression ratio of 43.81%. When cooled to low temperatures, its compressive strength increased to 196.12 N mm⁻², while the compression ratio decreased to 14.67%, corresponding to a 61.88% increase in compressive strength and a 66.51% decrease in compression ratio compared to room temperature. Conversely, at elevated temperatures, the propellant achieved a compressive strength of 103.92 N mm⁻² and a compression ratio of 69.86%, reflecting increases of 28.06% and 37.29%, respectively, relative to room temperature.

The study indicates that, based on comparisons, the propellant's compression ratio rises with increasing temperature, demonstrating enhanced compression responsiveness at higher temperatures. Additionally, the mechanical strength of the propellant exceeds that at room temperature under both high and low-temperature conditions. Although the experimental temperature range doesn't precisely replicate the operational temperatures the propellant encounters, there is a strong positive correlation between the findings. This implies that double-base propellants exhibit superior mechanical performance at temperature extremes.

4. Conclusions

The double-base propellant behaves like a typical pseudoplastic fluid, with its apparent viscosity decreasing significantly as shear rate increases. Through comprehensive rheological performance optimisation based on quantitative criteria (minimisation of K and τ₀, maximisation of n), the optimal process parameters for gelling the double-base propellant are determined to be: a gelation time of 80 minutes, a temperature of 50 °C, a solvent ratio (ethanol to acetone) of 2 : 3, and a solvent-to-NC ratio of 0.5.

The Ostwald-de Waele model best describes the shear-thinning behaviour, while the Herschel–Bulkley model captures the yield stress feature. The Arrhenius equation applies only at low shear rates (γ ≤ 20 s⁻¹). A constitutive function incorporating both temperature and shear rate: η(T, γ) = 7336.744 × exp(703.12/T) × γ^(0.2996 − 0.00372T),was established within the experimental parameter range.

Samples prepared under the optimised processing parameters were characterised for combustion, thermal, and mechanical performance. Combustion tests showed stable ignition with burn rates increasing with temperature. Thermal decomposition peak temperatures for NC/NG and nitroguanidine were 192.4 °C and 207.1 °C, with a mass loss of 92.5%. Mechanical properties were temperature-dependent, with compressibility increasing at higher temperatures.

A limitation of this study is that performance characterisation was conducted only on samples prepared under the optimised conditions. Future work should include comparative data from non-optimised samples to establish whether rheological optimisation itself enhances performance.

Author contributions

Jun Dong: writing – original draft, formal analysis. Yilong Duan: formal analysis, investigation. Changyong Mao: conceptualization. Jiyuan Dong: formal analysis. Xi He: investigation. Wei Ji: data curation. Jun Dong and Jiyuan Dong contributed equally to this work and are considered co-first authors. Corresponding authors: Changyong Mao and Wei Ji.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The datasets generated and analyzed in this current study are available from the corresponding author upon reasonable request.