Key Insights
The metal printing packaging market, valued at $54.96 million in 2025, is poised for robust growth, exhibiting a Compound Annual Growth Rate (CAGR) of 4.84% from 2025 to 2033. This expansion is fueled by several key drivers. The increasing demand for aesthetically appealing and durable packaging across diverse sectors like food & beverages, cosmetics, and pharmaceuticals is a significant catalyst. Furthermore, the growing preference for sustainable and recyclable packaging solutions is driving innovation within the metal printing packaging industry, leading to the adoption of eco-friendly printing techniques and materials. The shift towards personalized and branded packaging, along with advancements in digital printing technologies offering enhanced customization options, also contributes to market growth. However, the market faces challenges such as fluctuating raw material prices and concerns about the environmental impact of metal production and disposal. Despite these restraints, the market segmentation reveals significant opportunities. Offset lithography remains a dominant printing process, but flexography and digital printing are gaining traction due to their versatility and cost-effectiveness for shorter print runs. Major players like Ball Corporation, Crown Holdings Inc., and CCL Container are actively shaping the market landscape through strategic partnerships, technological advancements, and geographic expansion. The Asia-Pacific region is expected to witness substantial growth due to its burgeoning consumer base and increasing industrialization.
The competitive landscape is characterized by a mix of established players and emerging companies. Innovation in printing technologies, particularly in digital printing, is reshaping the industry, allowing for greater customization and shorter lead times. Companies are investing heavily in research and development to develop sustainable and efficient printing solutions, aligning with the growing consumer demand for environmentally friendly products. Regional variations in market dynamics are notable, with North America and Europe representing mature markets, while Asia-Pacific exhibits significant growth potential. The forecast period (2025-2033) anticipates continued expansion, driven by ongoing technological advancements, evolving consumer preferences, and the sustained growth of key end-use industries. A strategic focus on sustainable practices and customized solutions will be crucial for companies to capitalize on the market's growth potential.

Metal Printing Packaging Market: A Comprehensive Report (2019-2033)
This insightful report provides a detailed analysis of the Metal Printing Packaging Market, offering a comprehensive overview of its current state, future trajectory, and key players. The study period spans from 2019 to 2033, with 2025 serving as the base and estimated year. The forecast period covers 2025-2033, while the historical period encompasses 2019-2024. Expect in-depth analysis across key segments, including offset lithography, gravure, flexography, and digital printing technologies. Discover crucial market trends, challenges, and growth opportunities, empowering stakeholders to make informed decisions in this dynamic sector. The market is projected to reach xx Million by 2033, presenting significant potential for investors and businesses alike.
Metal Printing Packaging Market Market Composition & Trends
This section meticulously examines the Metal Printing Packaging Market's competitive landscape, encompassing market concentration, innovation drivers, regulatory frameworks, substitute products, end-user profiles, and mergers and acquisitions (M&A) activities. We delve into the market share distribution among key players like Ball Corporation, Crown Holdings Inc., CCL Container (CCL Industries Inc.), and others, providing a detailed understanding of the competitive dynamics at play. The report analyzes the impact of M&A activities, including deal values, on market consolidation and innovation. Furthermore, we assess the influence of evolving regulations, the emergence of substitute materials, and shifts in end-user demands on the market's overall trajectory.
- Market Concentration: Analysis of market share held by top players, highlighting the level of competition and potential for consolidation. xx% of the market is currently controlled by the top 5 players.
- Innovation Catalysts: Exploration of technological advancements driving innovation, such as advancements in digital printing and sustainable packaging solutions.
- Regulatory Landscape: Examination of relevant regulations impacting the industry, including those related to sustainability and food safety.
- Substitute Products: Assessment of alternative packaging materials and their competitive impact on metal printing packaging.
- End-User Profiles: Profiling of key end-user industries and their respective demands for metal printing packaging.
- M&A Activities: Analysis of recent mergers and acquisitions, along with their impact on market dynamics and future growth. Total M&A deal value for the past 5 years is estimated at xx Million.

Metal Printing Packaging Market Industry Evolution
This section provides a comprehensive analysis of the Metal Printing Packaging Market's growth trajectory, technological advancements, and shifting consumer preferences between 2019 and 2033. It meticulously tracks market growth rates, adoption rates of new technologies, and evolving consumer demands. We examine the impact of sustainability concerns, increasing demand for convenience, and the rise of e-commerce on market growth. The impact of the COVID-19 pandemic on supply chains and consumer behavior is also analyzed. The market is expected to exhibit a CAGR of xx% during the forecast period, driven by factors such as increased demand for sustainable packaging solutions and growing adoption of metal packaging in various end-use industries.
Leading Regions, Countries, or Segments in Metal Printing Packaging Market
This section identifies the dominant regions, countries, or segments within the Metal Printing Packaging Market, focusing on printing processes such as Offset Lithography, Gravure, Flexography, Digital, and Other Printing Technologies. We analyze the factors contributing to the dominance of specific regions or segments, considering investment trends, regulatory support, and market size.
- Key Drivers for the Dominant Segment (Example: Offset Lithography):
- High-volume printing capabilities and cost-effectiveness.
- Established infrastructure and widespread adoption across various industries.
- Mature technology with well-developed processes and expertise.
- In-depth analysis of dominance factors: Factors such as established infrastructure, high-volume production capabilities, and widespread industry adoption contribute significantly to the dominance of certain segments. The report provides detailed analysis to show what factors contribute to the dominance of the largest region and segment.
Metal Printing Packaging Market Product Innovations
This section details recent product innovations, applications, and performance metrics within the Metal Printing Packaging Market. We highlight unique selling propositions, technological advancements, and the impact of these innovations on market growth. Recent innovations include improved printing techniques that enhance product aesthetics and functionalities, as well as the development of sustainable and recyclable metal packaging solutions that meet the growing environmental consciousness of consumers and brands.
Propelling Factors for Metal Printing Packaging Market Growth
Several factors contribute to the growth of the Metal Printing Packaging Market. These include technological advancements in printing techniques and materials, increasing consumer demand for sustainable and eco-friendly packaging, and favorable government regulations promoting sustainable packaging solutions. The rising popularity of metal packaging in various industries, such as food and beverages, also fuels market growth.
Obstacles in the Metal Printing Packaging Market Market
The Metal Printing Packaging Market faces several challenges, including fluctuating raw material prices, stringent environmental regulations, and the competitive landscape featuring various packaging materials. Supply chain disruptions and economic downturns may hinder market expansion. These factors can impact market growth significantly. The high initial investment cost associated with establishing metal printing packaging production facilities may also create a barrier for some businesses.
Future Opportunities in Metal Printing Packaging Market
The Metal Printing Packaging Market presents promising opportunities for expansion. The increasing demand for sustainable and eco-friendly packaging creates a major avenue for growth. Technological advancements in printing and materials, coupled with increasing consumer awareness of sustainable products, offer significant opportunities for innovation and market expansion. Expanding into new markets and exploring new applications for metal packaging also present significant growth potential.
Major Players in the Metal Printing Packaging Market Ecosystem
- Ball Corporation
- Hubergroup Deutschland GmbH
- CCL Container (CCL Industries Inc.)
- Tonejet Limited
- Koenig & Bauer AG
- Crown Holdings Inc.
- Envases Metalurgicos de Alava SA
- Toyo Seihan Co Ltd (Toyo Seikan Group Holdings Ltd)
Key Developments in Metal Printing Packaging Market Industry
- May 2021: Ball Corporation launched its first-of-its-kind Ball Aluminum Cup, boosting sustainability and reducing plastic waste. The cup's availability in over 18,000 retailers across the U.S. significantly impacts market dynamics.
- March 2020: Bierland, a Brazilian craft beer brand, partnered with Crown Holdings Inc. to adopt metal can packaging for its entire product portfolio, highlighting the growing demand for sustainable alternatives.
Strategic Metal Printing Packaging Market Market Forecast
The Metal Printing Packaging Market is poised for substantial growth driven by technological advancements, rising consumer demand for sustainable packaging, and favorable government regulations. The forecast period indicates strong potential for market expansion, with key drivers expected to fuel significant increases in market value and volume. The market's robust future outlook is further bolstered by the ongoing development of innovative printing technologies and sustainable materials.
Metal Printing Packaging Market Segmentation
-
1. Printing Process
- 1.1. Offset Lithography
- 1.2. Gravure
- 1.3. Flexography
- 1.4. Digital
- 1.5. Other Printing Technologies
Metal Printing Packaging Market Segmentation By Geography
- 1. North America
- 2. Europe
- 3. Asia Pacific
- 4. Latin America
- 5. Middle East and Africa

Metal Printing Packaging Market REPORT HIGHLIGHTS
Aspects | Details |
---|---|
Study Period | 2019-2033 |
Base Year | 2024 |
Estimated Year | 2025 |
Forecast Period | 2025-2033 |
Historical Period | 2019-2024 |
Growth Rate | CAGR of 4.84% from 2019-2033 |
Segmentation |
|
Table of Contents
- 1. Introduction
- 1.1. Research Scope
- 1.2. Market Segmentation
- 1.3. Research Methodology
- 1.4. Definitions and Assumptions
- 2. Executive Summary
- 2.1. Introduction
- 3. Market Dynamics
- 3.1. Introduction
- 3.2. Market Drivers
- 3.2.1. Rapid Evolution of Digital Print Technology
- 3.3. Market Restrains
- 3.3.1. Monitoring issues and lack of standardization
- 3.4. Market Trends
- 3.4.1. Offset Lithography is Expected to Hold a Significant Market Growth
- 4. Market Factor Analysis
- 4.1. Porters Five Forces
- 4.2. Supply/Value Chain
- 4.3. PESTEL analysis
- 4.4. Market Entropy
- 4.5. Patent/Trademark Analysis
- 5. Global Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 5.1. Market Analysis, Insights and Forecast - by Printing Process
- 5.1.1. Offset Lithography
- 5.1.2. Gravure
- 5.1.3. Flexography
- 5.1.4. Digital
- 5.1.5. Other Printing Technologies
- 5.2. Market Analysis, Insights and Forecast - by Region
- 5.2.1. North America
- 5.2.2. Europe
- 5.2.3. Asia Pacific
- 5.2.4. Latin America
- 5.2.5. Middle East and Africa
- 5.1. Market Analysis, Insights and Forecast - by Printing Process
- 6. North America Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 6.1. Market Analysis, Insights and Forecast - by Printing Process
- 6.1.1. Offset Lithography
- 6.1.2. Gravure
- 6.1.3. Flexography
- 6.1.4. Digital
- 6.1.5. Other Printing Technologies
- 6.1. Market Analysis, Insights and Forecast - by Printing Process
- 7. Europe Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 7.1. Market Analysis, Insights and Forecast - by Printing Process
- 7.1.1. Offset Lithography
- 7.1.2. Gravure
- 7.1.3. Flexography
- 7.1.4. Digital
- 7.1.5. Other Printing Technologies
- 7.1. Market Analysis, Insights and Forecast - by Printing Process
- 8. Asia Pacific Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 8.1. Market Analysis, Insights and Forecast - by Printing Process
- 8.1.1. Offset Lithography
- 8.1.2. Gravure
- 8.1.3. Flexography
- 8.1.4. Digital
- 8.1.5. Other Printing Technologies
- 8.1. Market Analysis, Insights and Forecast - by Printing Process
- 9. Latin America Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 9.1. Market Analysis, Insights and Forecast - by Printing Process
- 9.1.1. Offset Lithography
- 9.1.2. Gravure
- 9.1.3. Flexography
- 9.1.4. Digital
- 9.1.5. Other Printing Technologies
- 9.1. Market Analysis, Insights and Forecast - by Printing Process
- 10. Middle East and Africa Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 10.1. Market Analysis, Insights and Forecast - by Printing Process
- 10.1.1. Offset Lithography
- 10.1.2. Gravure
- 10.1.3. Flexography
- 10.1.4. Digital
- 10.1.5. Other Printing Technologies
- 10.1. Market Analysis, Insights and Forecast - by Printing Process
- 11. North America Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 11.1. Market Analysis, Insights and Forecast - By Country/Sub-region
- 11.1.1.
- 12. Europe Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 12.1. Market Analysis, Insights and Forecast - By Country/Sub-region
- 12.1.1.
- 13. Asia Pacific Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 13.1. Market Analysis, Insights and Forecast - By Country/Sub-region
- 13.1.1.
- 14. Latin America Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 14.1. Market Analysis, Insights and Forecast - By Country/Sub-region
- 14.1.1.
- 15. Middle East and Africa Metal Printing Packaging Market Analysis, Insights and Forecast, 2019-2031
- 15.1. Market Analysis, Insights and Forecast - By Country/Sub-region
- 15.1.1.
- 16. Competitive Analysis
- 16.1. Global Market Share Analysis 2024
- 16.2. Company Profiles
- 16.2.1 Ball Corporation
- 16.2.1.1. Overview
- 16.2.1.2. Products
- 16.2.1.3. SWOT Analysis
- 16.2.1.4. Recent Developments
- 16.2.1.5. Financials (Based on Availability)
- 16.2.2 Hubergroup Deutschland GmbH
- 16.2.2.1. Overview
- 16.2.2.2. Products
- 16.2.2.3. SWOT Analysis
- 16.2.2.4. Recent Developments
- 16.2.2.5. Financials (Based on Availability)
- 16.2.3 CCL Container (CCL Industries Inc )
- 16.2.3.1. Overview
- 16.2.3.2. Products
- 16.2.3.3. SWOT Analysis
- 16.2.3.4. Recent Developments
- 16.2.3.5. Financials (Based on Availability)
- 16.2.4 Tonejet Limited
- 16.2.4.1. Overview
- 16.2.4.2. Products
- 16.2.4.3. SWOT Analysis
- 16.2.4.4. Recent Developments
- 16.2.4.5. Financials (Based on Availability)
- 16.2.5 Koenig & Bauer AG
- 16.2.5.1. Overview
- 16.2.5.2. Products
- 16.2.5.3. SWOT Analysis
- 16.2.5.4. Recent Developments
- 16.2.5.5. Financials (Based on Availability)
- 16.2.6 Crown Holdings Inc *List Not Exhaustive
- 16.2.6.1. Overview
- 16.2.6.2. Products
- 16.2.6.3. SWOT Analysis
- 16.2.6.4. Recent Developments
- 16.2.6.5. Financials (Based on Availability)
- 16.2.7 Envases Metalurgicos de Alava SA
- 16.2.7.1. Overview
- 16.2.7.2. Products
- 16.2.7.3. SWOT Analysis
- 16.2.7.4. Recent Developments
- 16.2.7.5. Financials (Based on Availability)
- 16.2.8 Toyo Seihan Co Ltd (Toyo Seikan Group Holdings Ltd)
- 16.2.8.1. Overview
- 16.2.8.2. Products
- 16.2.8.3. SWOT Analysis
- 16.2.8.4. Recent Developments
- 16.2.8.5. Financials (Based on Availability)
- 16.2.1 Ball Corporation
List of Figures
- Figure 1: Global Metal Printing Packaging Market Revenue Breakdown (Million, %) by Region 2024 & 2032
- Figure 2: North America Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 3: North America Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 4: Europe Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 5: Europe Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 6: Asia Pacific Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 7: Asia Pacific Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 8: Latin America Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 9: Latin America Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 10: Middle East and Africa Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 11: Middle East and Africa Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 12: North America Metal Printing Packaging Market Revenue (Million), by Printing Process 2024 & 2032
- Figure 13: North America Metal Printing Packaging Market Revenue Share (%), by Printing Process 2024 & 2032
- Figure 14: North America Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 15: North America Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 16: Europe Metal Printing Packaging Market Revenue (Million), by Printing Process 2024 & 2032
- Figure 17: Europe Metal Printing Packaging Market Revenue Share (%), by Printing Process 2024 & 2032
- Figure 18: Europe Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 19: Europe Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 20: Asia Pacific Metal Printing Packaging Market Revenue (Million), by Printing Process 2024 & 2032
- Figure 21: Asia Pacific Metal Printing Packaging Market Revenue Share (%), by Printing Process 2024 & 2032
- Figure 22: Asia Pacific Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 23: Asia Pacific Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 24: Latin America Metal Printing Packaging Market Revenue (Million), by Printing Process 2024 & 2032
- Figure 25: Latin America Metal Printing Packaging Market Revenue Share (%), by Printing Process 2024 & 2032
- Figure 26: Latin America Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 27: Latin America Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
- Figure 28: Middle East and Africa Metal Printing Packaging Market Revenue (Million), by Printing Process 2024 & 2032
- Figure 29: Middle East and Africa Metal Printing Packaging Market Revenue Share (%), by Printing Process 2024 & 2032
- Figure 30: Middle East and Africa Metal Printing Packaging Market Revenue (Million), by Country 2024 & 2032
- Figure 31: Middle East and Africa Metal Printing Packaging Market Revenue Share (%), by Country 2024 & 2032
List of Tables
- Table 1: Global Metal Printing Packaging Market Revenue Million Forecast, by Region 2019 & 2032
- Table 2: Global Metal Printing Packaging Market Revenue Million Forecast, by Printing Process 2019 & 2032
- Table 3: Global Metal Printing Packaging Market Revenue Million Forecast, by Region 2019 & 2032
- Table 4: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 5: Metal Printing Packaging Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 6: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 7: Metal Printing Packaging Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 8: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 9: Metal Printing Packaging Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 10: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 11: Metal Printing Packaging Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 12: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 13: Metal Printing Packaging Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 14: Global Metal Printing Packaging Market Revenue Million Forecast, by Printing Process 2019 & 2032
- Table 15: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 16: Global Metal Printing Packaging Market Revenue Million Forecast, by Printing Process 2019 & 2032
- Table 17: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 18: Global Metal Printing Packaging Market Revenue Million Forecast, by Printing Process 2019 & 2032
- Table 19: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 20: Global Metal Printing Packaging Market Revenue Million Forecast, by Printing Process 2019 & 2032
- Table 21: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
- Table 22: Global Metal Printing Packaging Market Revenue Million Forecast, by Printing Process 2019 & 2032
- Table 23: Global Metal Printing Packaging Market Revenue Million Forecast, by Country 2019 & 2032
Frequently Asked Questions
1. What is the projected Compound Annual Growth Rate (CAGR) of the Metal Printing Packaging Market?
The projected CAGR is approximately 4.84%.
2. Which companies are prominent players in the Metal Printing Packaging Market?
Key companies in the market include Ball Corporation, Hubergroup Deutschland GmbH, CCL Container (CCL Industries Inc ), Tonejet Limited, Koenig & Bauer AG, Crown Holdings Inc *List Not Exhaustive, Envases Metalurgicos de Alava SA, Toyo Seihan Co Ltd (Toyo Seikan Group Holdings Ltd).
3. What are the main segments of the Metal Printing Packaging Market?
The market segments include Printing Process.
4. Can you provide details about the market size?
The market size is estimated to be USD 54.96 Million as of 2022.
5. What are some drivers contributing to market growth?
Rapid Evolution of Digital Print Technology.
6. What are the notable trends driving market growth?
Offset Lithography is Expected to Hold a Significant Market Growth.
7. Are there any restraints impacting market growth?
Monitoring issues and lack of standardization.
8. Can you provide examples of recent developments in the market?
May 2021 - Ball Corporation launched a first-of-its-kind Ball Aluminum Cup that has the potential to advance sustainability and reduce plastic waste at gatherings irrespective of big and small. It will be available for purchase at major retailers in all 50 states across the U.S. Between May and June; the cups will be available for the first time in more than 18,000 food, drug, and mass retailers, including Kroger, Target, Albertsons, CVS, etc.
9. What pricing options are available for accessing the report?
Pricing options include single-user, multi-user, and enterprise licenses priced at USD 4750, USD 5250, and USD 8750 respectively.
10. Is the market size provided in terms of value or volume?
The market size is provided in terms of value, measured in Million.
11. Are there any specific market keywords associated with the report?
Yes, the market keyword associated with the report is "Metal Printing Packaging Market," which aids in identifying and referencing the specific market segment covered.
12. How do I determine which pricing option suits my needs best?
The pricing options vary based on user requirements and access needs. Individual users may opt for single-user licenses, while businesses requiring broader access may choose multi-user or enterprise licenses for cost-effective access to the report.
13. Are there any additional resources or data provided in the Metal Printing Packaging Market report?
While the report offers comprehensive insights, it's advisable to review the specific contents or supplementary materials provided to ascertain if additional resources or data are available.
14. How can I stay updated on further developments or reports in the Metal Printing Packaging Market?
To stay informed about further developments, trends, and reports in the Metal Printing Packaging Market, consider subscribing to industry newsletters, following relevant companies and organizations, or regularly checking reputable industry news sources and publications.
Methodology
Step 1 - Identification of Relevant Samples Size from Population Database



Step 2 - Approaches for Defining Global Market Size (Value, Volume* & Price*)

Note*: In applicable scenarios
Step 3 - Data Sources
Primary Research
- Web Analytics
- Survey Reports
- Research Institute
- Latest Research Reports
- Opinion Leaders
Secondary Research
- Annual Reports
- White Paper
- Latest Press Release
- Industry Association
- Paid Database
- Investor Presentations

Step 4 - Data Triangulation
Involves using different sources of information in order to increase the validity of a study
These sources are likely to be stakeholders in a program - participants, other researchers, program staff, other community members, and so on.
Then we put all data in single framework & apply various statistical tools to find out the dynamic on the market.
During the analysis stage, feedback from the stakeholder groups would be compared to determine areas of agreement as well as areas of divergence